Non-isomorphic Finite Groups Research Articles

On Spectra of Almost Simple Extensions of Even-Dimensional Orthogonal Groups

The spectrum of a finite group is the set of the orders of its elements. We consider the problem that arises within the framework of recognition of finite simple groups by spectrum: Determine all finite almost simple groups having the same spectrum as its socle. This problem was solved for all almost simple groups with exception of the case that the socle is a simple even-dimensional orthogonal group over a field of odd characteristic. Here we address this remaining case and determine the almost simple groups in question. Also we prove that there are infinitely many pairwise nonisomorphic finite groups having the same spectrum as the simple 8-dimensional symplectic group over a field of characteristic other than 7.

Several quantitative characterizations of some specific groups

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2,\ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1< p_2< \cdots < p_k$. The Gruenberg-Kegel graph of $G$, denoted $\operatorname{GK} (G)$, is defined as follows: its vertex set is $\pi(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i p_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots, d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi(G)$ is the vertex set of a connected component of $\operatorname{GK} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname{GK} (G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.

Finite groups acting symplectically on 𝑇²×𝑆²

For any symplectic form ω \omega on T 2 × S 2 T^2\times S^2 we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on T 2 × S 2 T^2\times S^2 that are trivial in cohomology but which do not admit any effective symplectic action on ( T 2 × S 2 , ω ) (T^2\times S^2,\omega ) . We also prove that for any ω \omega there is another symplectic form ω ′ \omega ’ on T 2 × S 2 T^2\times S^2 and a finite group acting symplectically and effectively on ( T 2 × S 2 , ω ′ ) (T^2\times S^2,\omega ’) which does not admit any effective symplectic action on ( T 2 × S 2 , ω ) (T^2\times S^2,\omega ) . A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of T 2 × S 2 T^2\times S^2 . A group G G is Jordan if there exists a constant C C such that any finite subgroup Γ \Gamma of G G contains an abelian subgroup whose index in Γ \Gamma is at most C C . Csikós, Pyber and Szabó proved recently that the diffeomorphism group of T 2 × S 2 T^2\times S^2 is not Jordan. We prove that, in contrast, for any symplectic form ω \omega on T 2 × S 2 T^2\times S^2 the group of symplectomorphisms S y m p ( T 2 × S 2 , ω ) \mathrm {Symp}(T^2\times S^2,\omega ) is Jordan. We also give upper and lower bounds for the optimal value of the constant C C in Jordan’s property for S y m p ( T 2 × S 2 , ω ) \mathrm {Symp}(T^2\times S^2,\omega ) depending on the cohomology class represented by ω \omega . Our bounds are sharp for a large class of symplectic forms on T 2 × S 2 T^2\times S^2 .

Groups with the same order and degree pattern

The degree pattern of a finite group has been introduced in [18]. A group M is called k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. It is shown that the alternating groups Am and Am+1, for m = 27, 35, 51, 57, 65, 77, 87, 93 and 95, are OD-characterizable, while their automorphism groups are 3-fold OD-characterizable. It is also shown that the symmetric groups Sm+2, for m = 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89 and 97, are 3-fold OD-characterizable. From this, the following theorem is derived. Let m be a natural number such that m ⩽ 100. Then one of the following holds: (a) if m ≠ 10, then the alternating groups Am are OD-characterizable, while the symmetric groups Sm are OD-characterizable or 3-fold OD-characterizable; (b) the alternating group A10 is 2-fold OD-characterizable; (c) the symmetric group S10 is 8-fold OD-characterizable. This theorem completes the study of OD-characterizability of the alternating and symmetric groups Am and Sm of degree m ⩽ 100.

More on the OD-Characterizability of a Finite Group

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut (O7(3)) and Aut (S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

OD-characterization of almost simple groups related to U 6(2)

Let G be a finite group and π( G)={p1, p2,…, pk} be the set of the primes dividing the order of G. We define its prime graph γ( G) as follows. The vertex set of this graph is π( G), and two distinct vertices p, q are joined by an edge if and only if pq ɛπ e( G). In this case, we write p∼q. For pɛπ( G), put deg(p):=|{qɛ π(G) | p∼ q|, which is called the degree of p. We also define D( G):=(deg(p1), deg(p2), …, deg(pk)), where p1 < p2 < … < pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L:=U 6(2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.

Recognizing alternating groups A p+3 for certain primes p by their orders and degree patterns

The degree pattern of a finite group M has been introduced by A. R. Moghaddamfar et al. [Algebra Colloquium, 2005, 12(3): 431–442]. A group M is called k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. In this article, we will show that the alternating groups Ap+3 for p = 23, 31, 37, 43 and 47 are OD-characterizable. Moreover, we show that the automorphism groups of these groups are 3-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.

OD-Characterization of Certain Finite Groups Having Connected Prime Graphs

The degree pattern of a finite group G denoted by D(G) was introduced in [5]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and same degree pattern as G. In the present article, we show that the alternating group A10 and the automorphism group Aut (McL) are 2-fold OD-characterizable, while the automorphism group Aut (J2) is 3-fold OD-characterizable and the symmetric group S10 is 8-fold OD-characterizable. It is worth mentioning that the prime graphs associated to these groups are connected and, in fact, among the groups with this property, they are the first groups which are investigated for OD-characterizability.

NOTES ON FINITE SIMPLE GROUPS WHOSE ORDERS HAVE THREE OR FOUR PRIME DIVISORS

Based on the prime graph of a finite group, its order can be divided into a product of some co-prime positive integers. These integers are called order components of this group. If there exist exactly k nonisomorphic finite groups with the same set of order components of a given finite group, we say that it is a k-recognizable group by its order component(s). In the present paper, we obtain that all finite simple Kn-groups (n = 3, 4) except U4(2) and A10can be uniquely determined by their order components. Moreover, U4(2) is 2-recognizable and A10is k-recognizable, where k denotes the number of all nonisomorphic classes of groups with the same order as A10. As a consequence of this result we can obtain some interesting corollaries.

Brauer t-tuples

Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. Generalizing the definition of a Brauer pair we introduce the notion of a Brauer t-tuple. We show that Brauer t-tuples exist for any t ⩾ 2 .

BRAUER PAIRS OF VZ-GROUPS

Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. A group is called a VZ-group if all its nonlinear irreducible characters vanish off the center. In this paper we give necessary and sufficient conditions for two non-isomorphic VZ-groups to form a Brauer pair.

On recognition of all finite nonabelian simple groups with orders having prime divisors at most 13

The spectrum of a group is the set of its element orders. We say that the problem of recognition by spectrum is solved for a finite group if we know the number of pairwise nonisomorphic finite groups with the same spectrum as the group under study. In this article the problem of recognition by spectrum is completely solved for every finite nonabelian simple group with orders having prime divisors at most 13.

Composition Factors from the Table of Marks

Let B(G) be the Burnside ring for a finite group G and let T(G) be the table of marks of G. It is known that, in general, neither B(G) nor T(G) determines the finite group G. That is, one can easily construct pairs of non-isomorphic finite groups such that their table of marks is the same (hence, their Burnside rings are isomorphic). In this note, we show that the composition factors of G are determined by the table of marks T(G).

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S9, S11, S12, S13, or A12, and h(ω(G))=2 if G is isomorphic to S2(6) or to O 8 + (2). 01

ON THE NUMBER OF FINITE GROUPS WITH A GIVEN AUTOMORPHISM GROUP

In this article, the following theorem is proved. Theorem. There exist only a finite number of pairwise nonisomorphic finite groups having at most a given number of automorphisms. Bibliography: 2 references.

Lacunary Fourier series on noncommutative groups

22akeinkX is called lacunary if nk+i/nk >X (> 1) for all k. This definition makes use of the ordering of the integers and does not immediately extend to two dimensions or to more general groups which have been recognized as a proper setting for large parts of Fourier Analysis. Let G be a compact group and as usual let G denote the set of equivalence classes of unitary irreducible representations of G. The set G has the following structure: To each pair a, PEG there is attached a measure p.a,/s on G. This is determined by the decomposition of the Kronecker product a(g>|8. In terms of this structure there is a natural duality between normal subgroups of G and certain subhypergroups of G. Some of the abelian Pontrjagin duality extends to this situation, although two nonisomorphic finite groups G may have the same hypergroup structure of G. The purpose of this note is to point out how, in certain instances, the hypergroup structure of G is related to properties of Fourier expansions on G. In particular we give a definition of a lacunary Fourier series on G in terms of G. If G is the circle group, our definition is formally quite different from the usual one but has similar implications. The definition is wide enough to cover the case of a series of the form 22ane<Xn, where xH are independent variables and a well known theorem of Kolmogoroff about such series can be extended to Fourier series on the product IX„ U(n), U(n) denoting the unitary group in n dimensions. Furthermore, the theorem of Banach stating that a lacunary Z^-series is an ZAseries is generalized to noncommutative groups. 2. The duality. We shall be concerned with compact groups G with normalized Haar measure dg and the familiar function spaces LX(G) and L2(G) of integrable and square integrable functions. The corresponding norms are denoted || ||i and || \\i. Every function/GL^G) can be uniquely represented by a Fourier series