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Involutions in the Automorphism Groups of Small Sporadic Simple Groups

Involutions in the Automorphism Groups of Small Sporadic Simple Groups Involutions in the Automorphism Groups of Small Sporadic Simple Groups div.banner_title_bkg div.trangle { border-color: #736C4D transparent transparent transparent; opacity:0.8; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=80)" ;filter: alpha(opacity=80); /*new styles end*/ } div.banner_title_bkg_if div.trangle { border-color: transparent transparent #736C4D transparent ; opacity:0.8; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=80)" ;filter: alpha(opacity=80); /*new styles end*/ } div.banner_title_bkg div.trangle { width: 90px; } #banner { background-image: url('http://images.hindawi.com/journals/algebra/algebra.banner.jpg'); background-position: 50% 0;} Hindawi Publishing Corporation Home Journals About Us Algebra About this Journal Submit a Manuscript Table of Contents Journal Menu About this Journal · Abstracting and Indexing · Advance Access · Aims and Scope · Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · Citations to this Journal · Contact Information · Editorial Board · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Open Special Issues · Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Algebra Volume 2015 (2015), Article ID 587629, 15 pages http://dx.doi.org/10.1155/2015/587629 Research Article Involutions in the Automorphism Groups of Small Sporadic Simple Groups Chris Bates , Peter Rowley , and Paul Taylor School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 6PL, UK Received 31 July 2014; Accepted 26 November 2014 Academic Editor: Peter Fleischmann Copyright © 2015 Chris Bates et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators. 1. Introduction Groups permeate many areas of mathematics. Sometimes they have cameo roles; other times they are centre stage. Frequently it is involutions, elements of order two, that are in the spotlight. For instance, in the topological arena we have involutory maps on the -sphere in connection with the Smith Conjecture (see, e.g., [ 1 – 3 ]), while in Banach spaces we encounter such things as involutive gradings and fixed points of involutions (see [ 4 , 5 ]). In areas of algebra, such as associative rings and algebraic groups, there are many sightings of involutions (see [ 6 – 8 ]). Involutions can often have a considerable influence on the structure of the group to which they belong. Even their absence can be telling—witness the Feit Thompson theorem [ 9 ]. For a finite group of even order, Brauer and Fowler [ 10 ] establish many results concerning involutions and other properties of the group. For example, they bound the index of a proper normal subgroup in terms of the number of involutions the group possesses. In a similar vein, for a finite group with at least two conjugacy classes of involutions the Thompson order formula ([ 11 ], Theorem 35.1) gives its order using data closely associated with the involutions. In the case when we have a finite nonabelian simple group, more often than not, its involutions play a dominant role (see, e.g., [ 12 ]). This paper studies the involutions in where is a small sporadic finite simple group. By small we mean that is isomorphic to one of the following groups: , , , , , , , , , , , , , , . The diminutive appellation aligns, more or less, with said group having a nontrivial permutation representation of degree at most 6156. Several of the larger sporadic groups have been studied individually in [ 13 – 16 ]. So for the remainder of this paper is assumed to be a small sporadic simple group and is a subgroup of containing . Also will denote an involution of . Put , the -conjugacy class of . Our aim is to study the suborbits of in its conjugation action on , or, in other words, to determine the action of on . This we do employing the services of the computational algebra packages Gap [ 17 ] and Magma [ 18 ] partnered by the electronic Atlas [ 19 ]. It goes without saying therefore that we use the Atlas notation and conventions as given in [ 20 ]. 2. Calculating Orbit Representatives As our starting point we take the smallest nontrivial permutation representation of as described in [ 19 ] with being generated by standard generators denoted here, as in [ 19 ], by and . In the case when , the standard generators for are, again as in [ 19 ], denoted by and . Having chosen a suitable element in to play the role of , we then forage for elements so that is a complete set of representatives for the -orbits of . In doing this we make frequent use of the standard command IsConjugate(H,x,y) —this works effectively here since the degree of the permutation representation of is no more than 6156. Also observe that we may write our conjugating element as a word in and . Thus, our aim is to find which have relatively “small” length relative to the generating set for . However we may not always achieve the minimum possible length. In more detail we proceed as follows: define and for For set The main purpose of is to produce a colony of short words in and , a number of which may well yield the same element of . To speed matters up we prune out these duplicates. We hunt through for typically at most 20, so as to ensnare suitable conjugating elements . If this does not yield enough representatives for the -orbits, then we recalculate with and replaced by short words in and which are also generators for . For example, we might try replacing by and keeping the same, or try a more complicated substitution such as replacing with . Such a substitution was used to produce a in the case when and in the class , starting from the word . Some of the words for the may be further simplified and this was done by hand. On a number of occasions this approach fails to deliver conjugating elements for some (often of small size) -orbits. To deal with such elusive -orbits, say , we begin by finding an in , usually by a random search. Then we obtain an element in such that . Now we run through some or all of the elements in , seeking a for which has small “length” (i.e., the total number of symbols in the given expression for ). Sometimes we may also vary . In somewhat more detail the Magma code employed is as shown in Algorithm 1 . Algorithm 1 We remark that this procedure is not usually as efficient as using the set and moreover usually yields more complicated expressions for the conjugating elements. We say a few words about how we guarantee that we have a representative of every orbit. By using the IsConjugate(Ct,x,y) command, where is a fixed involution of the -conjugacy class and Ct denotes , we ensure that no two representatives are in the same -orbit. We terminate our calculations for the -conjugacy class of when the sum of the -orbit sizes equals the size of the conjugacy class. One may envision the information presented here to be useful in the following circumstances. Suppose is a finite group with . Further suppose that we have identified the subgroup as being isomorphic to via recognizing and as standard generators. (See [ 21 ] for a discussion of standard generators.) Then we may translate our information from to so that we see how acts upon its involution conjugacy classes within the group . Moreover, if is large from a computational standpoint (e.g., is large or is a matrix group of large degree), then having the as short words may be beneficial. To facilitate applications such as these, computer files containing the as words in and are available on request from the second author. The following section gives the permutation ranks of on and the elements together with some additional information. 3. Orbit Representatives So is a suitably chosen (and then fixed) involution of the conjugacy class of . For a -conjugacy class we define Since is -invariant, will be a union of -orbits for each -conjugacy class for which . As is well known, may be calculated for any -conjugacy class using the complex character table (see, e.g., [ 22 ]), and this is easily carried out in Gap. As we proceed by breaking each (for ) into -orbits this is useful information. Our first table gives an overview of the permutation ranks of in its action on . The succeeding tables consider in turn the possibilities for , with the first column identifying and the second the size of the -orbit (contained in ) and the third supplies a group element for which . We emphasize that the following tables give the -orbits of —in the case when the tables are annotated so as to also yield the -orbits of . Before explaining how this is done, we remark that in all instances here when we have . Thus is a -conjugacy class and . Now suppose that and are two -conjugacy classes which fuse in (so is a -conjugacy class). Then for some and consequently . Hence, if is a -orbit contained in , will be a -orbit of . In this circumstance a broken horizontal line indicates that and are fused in and if is obtained by using the th listed conjugating element in then is obtained using the th listed conjugating element in . When a -conjugacy class is also a -conjugacy class, it may be the case that a -orbit contained in is the union of two -orbits in . A vertical line connecting two -orbits in signifies that their union is a -orbit. 3.1. Permutation Ranks See Table 1 . Table 1 3.2. 3.2.1. , See Table 2 . Table 2 3.3. 3.3.1. , See Table 3 . Table 3 3.3.2. , See Table 4 . Table 4 3.3.3. , See Table 5 . Table 5 3.4. 3.4.1. , See Table 6 . Table 6 3.5. 3.5.1. , See Table 7 . Table 7 3.5.2. , See Table 8 . Table 8 3.5.3. , See Table 9 . Table 9 3.6. 3.6.1. , See Table 10 . Table 10 3.6.2. , See Table 11 . Table 11 3.6.3. , See Table 12 . Table 12 3.7. 3.7.1. , See Table 13 . Table 13 3.8. 3.8.1. , See Table 14 . Table 14 3.8.2. , See Tables 15 and 16 . Table 15 Table 16 3.8.3. , See Table 17 . Table 17 3.8.4. , See Tables 18 and 19 . Table 18 Table 19 3.9. 3.9.1. , See Table 20 . Table 20 3.9.2. , See Table 21 . Table 21 3.10. 3.10.1. , See Tables 22 and 23 . Table 22 Table 23 3.10.2. , See Table 24 . Table 24 3.11. 3.11.1. , See Table 25 . Table 25 3.11.2. , See Table 26 . Table 26 3.12. 3.12.1. , See Table 27 . Table 27 3.12.2. , See Table 28 . Table 28 3.12.3. , See Table 29 . Table 29 3.13. 3.13.1. , See Table 30 . Table 30 3.13.2. , See Table 31 . Table 31 3.14. 3.14.1. , See Table 32 . Table 32 3.14.2. , See Table 33 . Table 33 3.14.3. , See Table 34 . Table 34 3.14.4. , See Table 35 . Table 35 3.15. 3.15.1. , See Tables 36 and 37 . Table 36 Table 37 3.15.2. , See Tables 38 and 39 . Table 38 Table 39 3.16. 3.16.1. , See Table 40 . Table 40 3.16.2. , See Table 41 . Table 41 3.16.3. , See Table 42 . Table 42 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References R. H. Bing, “A homeomorphism between the 3 -sphere and the sum of two solid horned spheres,” Annals of Mathematics: Second Series , vol. 56, pp. 354–362, 1952. View at Publisher · View at Google Scholar · View at MathSciNet D. Montgomery and L. Zippin, “Examples of transformation groups,” Proceedings of the American Mathematical Society , vol. 5, pp. 460–465, 1954. View at Publisher · View at Google Scholar · View at MathSciNet J. W. Morgan and H. Bass, Eds., The Smith Conjecture , vol. 112 of Pure and Applied Mathematics , Academic Press, Orlando, Fla, USA, 1984, Papers presented at the symposium held at Columbia University, New York, NY, USA, 1979. C. M. Edwards and G. T. Rüttimann, “Involutive and Peirce gradings in JBW ∗ -triples,” Communications in Algebra , vol. 31, no. 6, pp. 2819–2848, 2003. J. Gornicki and B. E. Rhoades, “A general fixed point theorem for involutions,” Indian Journal of Pure and Applied Mathematics , vol. 27, no. 1, pp. 13–23, 1996. View at MathSciNet · View at Scopus S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings , vol. 818 of Lecture Notes in Mathematics , Springer, Berlin, Germany, 1980. T. A. Springer, “The classification of involutions of simple algebraic groups,” Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics , vol. 34, no. 3, pp. 655–670, 1987. View at MathSciNet T. A. Springer, “Some results on algebraic group s with involutions,” in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983) , vol. 6 of Advanced Studies in Pure Mathematics , pp. 525–543, North-Holland, Amsterdam, The Netherlands, 1985. W. Feit and J. G. Thompson, “Solvability of groups of odd order,” Pacific Journal of Mathematics , vol. 13, pp. 775–1029, 1963. View at Publisher · View at Google Scholar · View at MathSciNet R. Brauer and K. A. Fowler, “On groups of even order,” Annals of Mathematics , vol. 62, no. 3, pp. 565–583, 1955. View at Publisher · View at Google Scholar · View at MathSciNet D. Gorenstein, The Classification of Finite Ssimple Groups. Vol. 1 , Groups of Noncharacteristic 2 Type. The University Series in Mathematics, Plenum Press, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups , vol. 40 of Mathematical Surveys and Monographs , American Mathematical Society, Providence, RI, USA, 1994. C. Bates and P. Rowley, “Involutions in Conway's largest simple group,” LMS Journal of Computation and Mathematics , vol. 7, pp. 337–351, 2004. View at Publisher · View at Google Scholar · View at MathSciNet P. Rowley and P. Taylor, “Point-line collinearity graphs of two sporadic minimal parabolic geometries,” Journal of Algebra , vol. 331, pp. 304–310, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus P. Rowley and P. Taylor, “Involutions in Janko's simple group J 4 ,” LMS Journal of Computation and Mathematics , vol. 14, pp. 238–253, 2011. View at Publisher · View at Google Scholar · View at MathSciNet P. Taylor, “Involutions in Fischer's sporadic groups,” preprint, http://eprints.ma.man.ac.uk/1622 . The GAP Group, GAP-Groups, Algorithms, Programming, Version 4.3 , 2002, http://www.gap-system.org . W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system. I. The user language,” Journal of Symbolic Computation , vol. 24, no. 3-4, pp. 235–265, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus R. Wilson, P. Walsh, J. Tripp et al., “Atlas o f finite group representations,” http://brauer.maths.qmul.ac.uk/Atlas/v3 . J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups , Clarendon, Oxford, UK, 1985. R. A. Wilson, “Standard generators for sporadic simple groups,” Journal of Algebra , vol. 184, no. 2, pp. 505–515, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus D. Gorenstein, Finite Groups , Chelsea Publishing, New York, NY, USA, 2nd edition, 1980. View at MathSciNet (function (i, s, o, g, r, a, m) { i['GoogleAnalyticsObject'] = r; i[r] = i[r] || function () { (i[r].q = i[r].q || []).push(arguments) }, i[r].l = 1 * new Date(); a = s.createElement(o), m = s.getElementsByTagName(o)[0]; a.async = 1; a.src = g; m.parentNode.insertBefore(a, m) })(window, document, 'script', '//www.google-analytics.com/analytics.js', 'ga'); ga('create', 'UA-8578054-2', 'auto'); ga('send', 'pageview'); http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Hindawi Publishing Corporation

Involutions in the Automorphism Groups of Small Sporadic Simple Groups

Bates, Chris; Rowley, Peter; Taylor, Paul
Algebra , Volume 2015 (2015) – Feb 28, 2015
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Involutions in the Automorphism Groups of Small Sporadic Simple Groups div.banner_title_bkg div.trangle { border-color: #736C4D transparent transparent transparent; opacity:0.8; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=80)" ;filter: alpha(opacity=80); /*new styles end*/ } div.banner_title_bkg_if div.trangle { border-color: transparent transparent #736C4D transparent ; opacity:0.8; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=80)" ;filter: alpha(opacity=80); /*new styles end*/ } div.banner_title_bkg div.trangle { width: 90px; } #banner { background-image: url('http://images.hindawi.com/journals/algebra/algebra.banner.jpg'); background-position: 50% 0;} Hindawi Publishing Corporation Home Journals About Us Algebra About this Journal Submit a Manuscript Table of Contents Journal Menu About this Journal · Abstracting and Indexing · Advance Access · Aims and Scope · Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · Citations to this Journal · Contact Information · Editorial Board · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Open Special Issues · Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Algebra Volume 2015 (2015), Article ID 587629, 15 pages http://dx.doi.org/10.1155/2015/587629 Research Article Involutions in the Automorphism Groups of Small Sporadic Simple Groups Chris Bates , Peter Rowley , and Paul Taylor School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 6PL, UK Received 31 July 2014; Accepted 26 November 2014 Academic Editor: Peter Fleischmann Copyright © 2015 Chris Bates et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators. 1. Introduction Groups permeate many areas of mathematics. Sometimes they have cameo roles; other times they are centre stage. Frequently it is involutions, elements of order two, that are in the spotlight. For instance, in the topological arena we have involutory maps on the -sphere in connection with the Smith Conjecture (see, e.g., [ 1 – 3 ]), while in Banach spaces we encounter such things as involutive gradings and fixed points of involutions (see [ 4 , 5 ]). In areas of algebra, such as associative rings and algebraic groups, there are many sightings of involutions (see [ 6 – 8 ]). Involutions can often have a considerable influence on the structure of the group to which they belong. Even their absence can be telling—witness the Feit Thompson theorem [ 9 ]. For a finite group of even order, Brauer and Fowler [ 10 ] establish many results concerning involutions and other properties of the group. For example, they bound the index of a proper normal subgroup in terms of the number of involutions the group possesses. In a similar vein, for a finite group with at least two conjugacy classes of involutions the Thompson order formula ([ 11 ], Theorem 35.1) gives its order using data closely associated with the involutions. In the case when we have a finite nonabelian simple group, more often than not, its involutions play a dominant role (see, e.g., [ 12 ]). This paper studies the involutions in where is a small sporadic finite simple group. By small we mean that is isomorphic to one of the following groups: , , , , , , , , , , , , , , . The diminutive appellation aligns, more or less, with said group having a nontrivial permutation representation of degree at most 6156. Several of the larger sporadic groups have been studied individually in [ 13 – 16 ]. So for the remainder of this paper is assumed to be a small sporadic simple group and is a subgroup of containing . Also will denote an involution of . Put , the -conjugacy class of . Our aim is to study the suborbits of in its conjugation action on , or, in other words, to determine the action of on . This we do employing the services of the computational algebra packages Gap [ 17 ] and Magma [ 18 ] partnered by the electronic Atlas [ 19 ]. It goes without saying therefore that we use the Atlas notation and conventions as given in [ 20 ]. 2. Calculating Orbit Representatives As our starting point we take the smallest nontrivial permutation representation of as described in [ 19 ] with being generated by standard generators denoted here, as in [ 19 ], by and . In the case when , the standard generators for are, again as in [ 19 ], denoted by and . Having chosen a suitable element in to play the role of , we then forage for elements so that is a complete set of representatives for the -orbits of . In doing this we make frequent use of the standard command IsConjugate(H,x,y) —this works effectively here since the degree of the permutation representation of is no more than 6156. Also observe that we may write our conjugating element as a word in and . Thus, our aim is to find which have relatively “small” length relative to the generating set for . However we may not always achieve the minimum possible length. In more detail we proceed as follows: define and for For set The main purpose of is to produce a colony of short words in and , a number of which may well yield the same element of . To speed matters up we prune out these duplicates. We hunt through for typically at most 20, so as to ensnare suitable conjugating elements . If this does not yield enough representatives for the -orbits, then we recalculate with and replaced by short words in and which are also generators for . For example, we might try replacing by and keeping the same, or try a more complicated substitution such as replacing with . Such a substitution was used to produce a in the case when and in the class , starting from the word . Some of the words for the may be further simplified and this was done by hand. On a number of occasions this approach fails to deliver conjugating elements for some (often of small size) -orbits. To deal with such elusive -orbits, say , we begin by finding an in , usually by a random search. Then we obtain an element in such that . Now we run through some or all of the elements in , seeking a for which has small “length” (i.e., the total number of symbols in the given expression for ). Sometimes we may also vary . In somewhat more detail the Magma code employed is as shown in Algorithm 1 . Algorithm 1 We remark that this procedure is not usually as efficient as using the set and moreover usually yields more complicated expressions for the conjugating elements. We say a few words about how we guarantee that we have a representative of every orbit. By using the IsConjugate(Ct,x,y) command, where is a fixed involution of the -conjugacy class and Ct denotes , we ensure that no two representatives are in the same -orbit. We terminate our calculations for the -conjugacy class of when the sum of the -orbit sizes equals the size of the conjugacy class. One may envision the information presented here to be useful in the following circumstances. Suppose is a finite group with . Further suppose that we have identified the subgroup as being isomorphic to via recognizing and as standard generators. (See [ 21 ] for a discussion of standard generators.) Then we may translate our information from to so that we see how acts upon its involution conjugacy classes within the group . Moreover, if is large from a computational standpoint (e.g., is large or is a matrix group of large degree), then having the as short words may be beneficial. To facilitate applications such as these, computer files containing the as words in and are available on request from the second author. The following section gives the permutation ranks of on and the elements together with some additional information. 3. Orbit Representatives So is a suitably chosen (and then fixed) involution of the conjugacy class of . For a -conjugacy class we define Since is -invariant, will be a union of -orbits for each -conjugacy class for which . As is well known, may be calculated for any -conjugacy class using the complex character table (see, e.g., [ 22 ]), and this is easily carried out in Gap. As we proceed by breaking each (for ) into -orbits this is useful information. Our first table gives an overview of the permutation ranks of in its action on . The succeeding tables consider in turn the possibilities for , with the first column identifying and the second the size of the -orbit (contained in ) and the third supplies a group element for which . We emphasize that the following tables give the -orbits of —in the case when the tables are annotated so as to also yield the -orbits of . Before explaining how this is done, we remark that in all instances here when we have . Thus is a -conjugacy class and . Now suppose that and are two -conjugacy classes which fuse in (so is a -conjugacy class). Then for some and consequently . Hence, if is a -orbit contained in , will be a -orbit of . In this circumstance a broken horizontal line indicates that and are fused in and if is obtained by using the th listed conjugating element in then is obtained using the th listed conjugating element in . When a -conjugacy class is also a -conjugacy class, it may be the case that a -orbit contained in is the union of two -orbits in . A vertical line connecting two -orbits in signifies that their union is a -orbit. 3.1. Permutation Ranks See Table 1 . Table 1 3.2. 3.2.1. , See Table 2 . Table 2 3.3. 3.3.1. , See Table 3 . Table 3 3.3.2. , See Table 4 . Table 4 3.3.3. , See Table 5 . Table 5 3.4. 3.4.1. , See Table 6 . Table 6 3.5. 3.5.1. , See Table 7 . Table 7 3.5.2. , See Table 8 . Table 8 3.5.3. , See Table 9 . Table 9 3.6. 3.6.1. , See Table 10 . Table 10 3.6.2. , See Table 11 . Table 11 3.6.3. , See Table 12 . Table 12 3.7. 3.7.1. , See Table 13 . Table 13 3.8. 3.8.1. , See Table 14 . Table 14 3.8.2. , See Tables 15 and 16 . Table 15 Table 16 3.8.3. , See Table 17 . Table 17 3.8.4. , See Tables 18 and 19 . Table 18 Table 19 3.9. 3.9.1. , See Table 20 . Table 20 3.9.2. , See Table 21 . Table 21 3.10. 3.10.1. , See Tables 22 and 23 . Table 22 Table 23 3.10.2. , See Table 24 . Table 24 3.11. 3.11.1. , See Table 25 . Table 25 3.11.2. , See Table 26 . Table 26 3.12. 3.12.1. , See Table 27 . Table 27 3.12.2. , See Table 28 . Table 28 3.12.3. , See Table 29 . Table 29 3.13. 3.13.1. , See Table 30 . Table 30 3.13.2. , See Table 31 . Table 31 3.14. 3.14.1. , See Table 32 . Table 32 3.14.2. , See Table 33 . Table 33 3.14.3. , See Table 34 . Table 34 3.14.4. , See Table 35 . Table 35 3.15. 3.15.1. , See Tables 36 and 37 . Table 36 Table 37 3.15.2. , See Tables 38 and 39 . Table 38 Table 39 3.16. 3.16.1. , See Table 40 . Table 40 3.16.2. , See Table 41 . Table 41 3.16.3. , See Table 42 . Table 42 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References R. H. Bing, “A homeomorphism between the 3 -sphere and the sum of two solid horned spheres,” Annals of Mathematics: Second Series , vol. 56, pp. 354–362, 1952. View at Publisher · View at Google Scholar · View at MathSciNet D. Montgomery and L. Zippin, “Examples of transformation groups,” Proceedings of the American Mathematical Society , vol. 5, pp. 460–465, 1954. View at Publisher · View at Google Scholar · View at MathSciNet J. W. Morgan and H. Bass, Eds., The Smith Conjecture , vol. 112 of Pure and Applied Mathematics , Academic Press, Orlando, Fla, USA, 1984, Papers presented at the symposium held at Columbia University, New York, NY, USA, 1979. C. M. Edwards and G. T. Rüttimann, “Involutive and Peirce gradings in JBW ∗ -triples,” Communications in Algebra , vol. 31, no. 6, pp. 2819–2848, 2003. J. Gornicki and B. E. Rhoades, “A general fixed point theorem for involutions,” Indian Journal of Pure and Applied Mathematics , vol. 27, no. 1, pp. 13–23, 1996. View at MathSciNet · View at Scopus S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings , vol. 818 of Lecture Notes in Mathematics , Springer, Berlin, Germany, 1980. T. A. Springer, “The classification of involutions of simple algebraic groups,” Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics , vol. 34, no. 3, pp. 655–670, 1987. View at MathSciNet T. A. Springer, “Some results on algebraic group s with involutions,” in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983) , vol. 6 of Advanced Studies in Pure Mathematics , pp. 525–543, North-Holland, Amsterdam, The Netherlands, 1985. W. Feit and J. G. Thompson, “Solvability of groups of odd order,” Pacific Journal of Mathematics , vol. 13, pp. 775–1029, 1963. View at Publisher · View at Google Scholar · View at MathSciNet R. Brauer and K. A. Fowler, “On groups of even order,” Annals of Mathematics , vol. 62, no. 3, pp. 565–583, 1955. View at Publisher · View at Google Scholar · View at MathSciNet D. Gorenstein, The Classification of Finite Ssimple Groups. Vol. 1 , Groups of Noncharacteristic 2 Type. The University Series in Mathematics, Plenum Press, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups , vol. 40 of Mathematical Surveys and Monographs , American Mathematical Society, Providence, RI, USA, 1994. C. Bates and P. Rowley, “Involutions in Conway's largest simple group,” LMS Journal of Computation and Mathematics , vol. 7, pp. 337–351, 2004. View at Publisher · View at Google Scholar · View at MathSciNet P. Rowley and P. Taylor, “Point-line collinearity graphs of two sporadic minimal parabolic geometries,” Journal of Algebra , vol. 331, pp. 304–310, 2011. 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