Centralizer Of Involution Research Articles

Regular polyhedra related to projective linear groups

For each odd prime p, there is a regular polyhedron Π p of type {3, p} with 1 2 ( p 2−1) vertices whose rotation group is PSL(2, p); its complete group is PSL(2, p) × Z 2 or PGL(2, p) as p ≡ 1 or 3 (mod 4). If p ≡ 1 (mod 4), then the group of Π p contains a central involution, and identification of antipodal vertices under this involution yields another regular polyhedron Π p 2 of type {3, p} with 1 4 ( p 2)−1) vertices and group PSL(2, p). Realizations of the polyhedra in euclidean spaces are briefly described.

A character-theory-free characterization of the Mathieu group M12

AbstractThe known characterization of the Mathieu group M12 by the structure of the centralizer of a 2-central involution is based on the application of the theory of exceptional characters and uses in addition a block theoretic result which asserts that a simple group of order |M12| is isomorphic to M12. The details of the proof of the latter result had never been published. We show here that M12 can be handled in a completely elementary and group theoretical way.

Characterization of infinite Chernikov groups

It is shown that an infinite locally finite group is a Chernikov group if and only if its Cartesian square G × G contains a subgroup T of finite index such that Aut T possesses the four group V as subgroup with Chernikov centralizer CT(v) and the centralizers of involutions in V are weakly isolated in T.

Turing machine computations in finitely axiomatizable theories

LITERATURE CITED I. B. M. Veretennikov and A. A. Makhnev, "Finite groups with restricted centralizer of involution," Dep. VINITI No. 1046-81. 2. A. S. Konrat'ev, "Finite simple groups, whose Sylow 2-subgroup is an extension of an Abelian group by a group of rank i," Algebra Logika, 14, No. 3, 288-303 (1975). 3. A. A. Makhnev, "Finite groups with centralizer of order 6," Algebra Logika, 16, No. 4, 432-442 (1977). 4. A. A. Makhnev, "Generation of finite groups by classes of involutions," Mat. Sb., iii, No. 2, 266-278 (1980). 5. S. A. Syskin,"Centralizers of 2-subgroups in finite groups," Algebra Logika, 17, No. 3, 316-354 (1978). 6. N. K. Dickson, "Groups with dihedral 3-normalizers of order 4k, i," J. Algebra, 54, No. 2, 390-409 (1978). 7. D. M. Goldschmidt. "2-fusion in finite groups," Ann. Math., 99, No. i, 70-117 (1974). 8. R. L. Griess, D. R. Mason, and G. M. Seitz, "Bender groups as standard subgroups," Trans. Am. Math. Soc., 238, 179-212 (1978). 9. G. Higman, "Odd characterizations of finite simple groups," Univ. of Michigan (1968). i0. J. H. Walter, "The characterization of finite groups with abelian Sylow 2-subgroups," Ann. Math., 89, No. 3, 405-514 (1969).

Finite groups having an involution centralizer with a PSU(3, 3) component

Finite groups having an involution centralizer with a PSU(3, 3) component

Characterizations of the Fischer Groups. I, II, III

B. Fischer, in his work on finite groups which contain a conjugacy class of $3$-transpositions, discovered three new sporadic finite simple groups, usually denoted $M(22)$, $M(23)$ and $M(24)’$. In Part I two of these groups, $M(22)$ and $M(23)$, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups ${U_6}(2)$ (often denoted by $M(21))$ and $P\Omega (7,3)$, both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups $M(24)$ is not simple but contains a simple subgroup $M(24)’$ of index two. In Part II we give a similar characterization by the centralizer of a central involution of $M(24)$ and also a partial characterization of the simple group $M(24)’$. The purpose of Part III is to complete the characterization of $M(24)’$ by showing that our abstract group $G$ is isomorphic to $M(24)’$. We first prove that $G$ contains a subgroup $X \cong M(23)$ and then we construct a graph (on the cosets of $X$) which is shown to be isomorphic to the graph for $M(24)$.

From a 3-local plus 3-fusion to the centralizer of an involution

It is shown that much of the structure of the centralizer of a central involution in a group of characteristic 2 type with a standard 3 3 -component of type G L ( n , 2 ) GL(n,2) , n ⩾ 6 n \geqslant 6 , is easily determined from the 3 3 -fusion. Consequently, one can shorten the previous treatment of such groups by Finkelstein and the author.

Characterizations of the Fischer groups. I, II, III

B. Fischer, in his work on finite groups which contain a conjugacy class of 3 3 -transpositions, discovered three new sporadic finite simple groups, usually denoted M ( 22 ) M(22) , M ( 23 ) M(23) and M ( 24 ) ′ M(24)’ . In Part I two of these groups, M ( 22 ) M(22) and M ( 23 ) M(23) , are characterized by the structure of the centralizer of a central involution. In addition, the simple groups U 6 ( 2 ) {U_6}(2) (often denoted by M ( 21 ) ) M(21)) and P Ω ( 7 , 3 ) P\Omega (7,3) , both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups M ( 24 ) M(24) is not simple but contains a simple subgroup M ( 24 ) ′ M(24)’ of index two. In Part II we give a similar characterization by the centralizer of a central involution of M ( 24 ) M(24) and also a partial characterization of the simple group M ( 24 ) ′ M(24)’ . The purpose of Part III is to complete the characterization of M ( 24 ) ′ M(24)’ by showing that our abstract group G G is isomorphic to M ( 24 ) ′ M(24)’ . We first prove that G G contains a subgroup X ≅ M ( 23 ) X \cong M(23) and then we construct a graph (on the cosets of X X ) which is shown to be isomorphic to the graph for M ( 24 ) M(24) .

Finite groups with given centralizer of a central involution

Finite groups with given centralizer of a central involution

Finite groups having an involution centralizer with a 2-component of type PSL(3,3)

Finite groups having an involution centralizer with a 2-component of type PSL(3<i>,</i>3)

On centralizers of involutions in 2-groups

1. Introduction. It is well known (see (5) and (6)) that if T is a 2-group containing an involution t such that |CT(t)| = 4, then T is dihedral, semidihedral or cyclic of order 4. Fomin(1) has studied 2-groups T containing an involution t such that |CT(t)| ≤ 8. He showed in particular that such a group has sectional 2-rank at most 4. In this paper we will prove the following:Theorem A. Let T be a 2-group containing an involution t such that |CT(t)| ≤ 16. Then T has sectional 2-rank at most 6.

Hamiltonian Cubic Graphs and Centralizers of Involutions

In 1948, R. Frucht [5] proved that, given a finite group G, there are infinitely many connected cubic graphs X such that the automorphism group Aut X is isomorphic to G. In a letter, Professor Frucht has proposed the problem, whether in addition X can be required to be hamiltonian. One of the aims of the present note is to answer this question affirmatively.THEOREM 1.1. Given a finite group G there are infinitely many finite hamiltonian cubic graphs Y such that Aut Y ≌ G.In fact, we prove the following:THEOREM 1.2. Given a finite cubic graph X having no component isomorphic to K4, there exists a hamiltonian cubic graph Y such that Aut Y ≌ Aut X and |V(Y)| = 6|V(X)|.This implies 1.1 by the theorem of Frucht [5] mentioned above.

On transitive permutation groups in which a 2-central involution fixes a unique point

On transitive permutation groups in which a 2-central involution fixes a unique point

Ranks of intersections of centralizers of involutions in finite groups

Ranks of intersections of centralizers of involutions in finite groups

Finite groups having an involution centralizer with a $2$-component of dihedral type, I

Finite groups having an involution centralizer with a $2$-component of dihedral type, I

Finite groups having an involution centralizer with a $2$-component of dihedral type, II

Finite groups having an involution centralizer with a $2$-component of dihedral type, II

On centralizers of involutions with components of 2-rank two, I

THEOREM A. Let G be a jInite non-abelian simple group containing an involution t such that for H0 = Co(t) and H = H,/O(H,,), the following assumptions hold: H contain a nonsokable component A such that m(A/Z(A)) = 2, r(A/Z(A)) = 3, m(Cff(A)) = 1. Further assume that one of the following conditions holds: (i) The order of A is not divisible by 28 and H = AC,(A). (ii) The order of C,(A) is 2 and H = AC,(A). (iii) The order of G is not divisible by 211. Then G is isomorphic to one of the following groups:

On centralizers of involutions with components of 2-rank two, II

THEOREM A. Let G be a jInite non-abelian simple group containing an involution t such that for H0 = Co(t) and H = H,/O(H,,), the following assumptions hold: H contain a nonsokable component A such that m(A/Z(A)) = 2, r(A/Z(A)) = 3, m(Cff(A)) = 1. Further assume that one of the following conditions holds: (i) The order of A is not divisible by 28 and H = AC,(A). (ii) The order of C,(A) is 2 and H = AC,(A). (iii) The order of G is not divisible by 211. Then G is isomorphic to one of the following groups:

Decomposable involution centralizers involving exceptional lie type simple groups

There have been investigations (Janko (1966), Janko and Thompson (1966), Yamaki (1972)) of finite groups G which contain a central involution t whose centralizer (in G) has the form C(t) = 〈t〉 × F, where F is isomorphic to a non-abelian simple group. Here it is shown such a group cannot be simple when F is isomorphic to an exceptional Lie type simple group of odd characteristic. Specifically the following theorem is proved.

A 2-local characterization of the simple group E

The purpose of this paper is to characterize the simple group E, [6] by a particular 2-local subgroup. First, some properties of a Sylow 2-subgroup of the 2-local subgroup will be studied. Second, we will show that a Sylow 2-subgroup of the 2-local subgroup is a Sylow 2-subgroup of the simple group in question. Finally, we will exhibit the centralizer of an involution and quote a result of Parrott [5] that characterizes the group E by the centralizer of an involution. The 2-local subgroup is the normalizer of an elementary abelian subgroup V of order 32. The normalizer of V is the (unique) nonsplit extension of 17 by GL(5, 2). This extension was shown to exist by Thompson [7], and its uniqueness was shown by Dempwolff [l]. For notation, let G be a simple group containing N = :V,( V) z 25GL(5, 2) as a 2-local subgroup. The centralizer of a central involution T+ in N is a group C,(v,) such that C,-(v~)/O,(C,(V,)) _c A,, with O,(C,(v,)) isomorphic to an extra-special group of order 29 of + type. The precise statement is: