Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

Conjugate biprimitive finite groups saturated with finite simple subgroupsU 3(2 n )

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U₃(2ⁿ)}. It is proved that every such group is isomorphic to a simple group U₃(Q) over a locally finite field Q of characteristic 2. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998.     

Periodic part in some Shunkov groups

A group G is saturated with groups in a set X if every finite subgroup of G is embeddable in G into a subgroup L isomorphic to some group in X. We show that a Shunkov group has a periodic part if the saturating set for it coincides with one of the following: {L₂(q)}, {Sz(q)}, {Re(q)}, or {U₃(2ⁿ)}. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 96–125, January–February, 1999.     

Conjugate biprimitive finite groups saturated with finite simple subgroups

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L₂(pⁿ)}, {Re(3²ⁿ⁺¹)}, and {Sz(2²ⁿ⁺¹)}. It is proved thai such groups are all isomorphic to {L₂(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 224–245, March–April, 1998.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Quasirecognizability by the Set of Element Orders for Groups 3 D 4(q), for q Even

It is proved that if G is a finite group with an element order set as in the simple group ³D₄(q), where q is even, then the commutant of G/F(G) is isomorphic to ³D₄(q) and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3–19, January–February, 2006.     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

The intersection of Sylow subgroups in finite groups

In Theorem 1, letting p be a prime, we prove: (1) If G=S_n is a symmetric group of degree n, then G contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 2), (2, 4), (2, 8)}, and (2) If H=A_n is an alternating group of degree n, then H contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 4)}. In Theorem 2, we argue that if G is a finite simple non-Abelian group and p is a prime, then G contains a pair of Sylow p-subgroups with trivial intersection. Also we present the corollary which says that if P is a Sylow subgroup of a finite simple non-Abelian group G, then ‖G‖>‖P‖². Supported by RFFR grants Nos. 93-01-01529, 93-01-01501, and 96-01-01893, and by International Science Foundation and Government of Russia grant RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 424–432, July–August, 1996.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

Element orders in coverings of symmetric and alternating groups

We prove that if the factor group H=G/N of a finite group G is isomorphic to a symmetric or alternating group of degree m, where m≥5 and N≠1, then G has an element whose order is distinct from any element’s order in H. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 296–315, May–June, 1999.     

On two classes of permutations with number-theoretic conditions on the lengths of the cycles

Let Λ be an arbitrary set of positive integers andS _n (Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |S _n (Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |S _n (Λ)| stands for the number of elements in the finite setS _n (Λ). Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 881–891, December, 1997. Translated by A. I. Shtern     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26. Supported by RFBR (project Nos. 08-01-00322 and 06-01-39001), by SB RAS (Integration Project No. 2006.1.2), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and Young Doctors and Candidates of Science (grants MD-2848.2007.1 and MK-377.2008.1). Translated from Algebra i Logika, Vol. 47, No. 5, pp. 558–570, September–October, 2008.