CC BY
32
MSC 20D10, 20D25, 20F16
On the structure of finite simply reducible
groups
© L. S. Kazarin, E. I. Chankov
Demidov Yaroslavl State University, Yaroslavl, Russia
A group G is called simply reducible (SR-group) if every character of G is real valued and the tensor product of any two irreducible representations of G is multiplicity free. The main result states the following: Let G be a finite simply reducible group; then G is solvable. Also we determine the structure of finite supersolvable SR-groups and discuss some other properties of SR-groups
Keywords: finite groups, tensor products, representations, solvability
1. Introduction
Definition 1 A group G is called simply reducible (briefly, a SE-group) if it has following two properties:
(1) every element of this group is conjugate to its inverse;
(2) the tensor product of any two irreducible ordinary representations decomposes into a sum of irreducible representations of the group G with multiplicities not exceeding 1.
The class of SE-groups was introduced by E. P. Wigner [1]. Condition (2) in the definition is essential for physical applications. This condition implies that «regular linear combinations» of products of basis functions are determined up to a phase factor and that a solution of a physical problem is uniquely determined from symmetry considerations (see [2]). A certain generalization of these groups was proposed by G.W. Markov [3]. The problem of description of SE-groups was also pointed out by A. I. Kostrikin in the supplement «Unsolved problems» of the book [4]. S. P. Strunkov posed the question of the solvability of finite simply reducible groups ([5], Problem 11.94). J. Saxl formulated the problem of finding all finite groups with the following property: the tensor square of any irreducible ordinary representation of the group is multiplicity free ([5], Problem 9.56).
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Wigner’s condition. E, P, Wigner proved for any finite group G the following inequality:
' Eivgi3 < E iCo(g)i2,
geo geO
where *jg = {x G G i x2 = g} and Co (g) is the centralizer of the element g. For a group G the above nequalitv becomes an equality if and only if G is an SR-group,
We note the following result by G, W, Markov [6]:
Let G be a finite group and let K be the diagonal subgroup in H = G x G x G, Then G is simply reducible if and only if every (K, K) double coset in H is self inverse. In other words G is a SR-group if and only if (H, K) is a svmetrie Gel’fand pair. In particular, Wigner’s inequality can be obtained from Mackey’s result,
G
and x is an irreducible character of G, then x(1) < k(G) and |G| < k(G)3, where k(G) denotes the number of the conjugacy elasses of the group G, Using this result, they reduced the question of the solvability of a finite simply reducible group to the specific case where non-Abelian composition factors of the group belong to alternating groups of degree at most 6,
§ 2. Finite simply reducible groups are solvable
We have obtained [8] an affirmative solution of the problem posed by S, P. Strunkov concerning the solvability of finite simply reducible groups. The following definition was proposed in [7].
G
irreducible ordinary representation of this group decomposes into the sum of irreducible representations of G with multiplicities not exceeding 1,
The weakening is achieved both by eliminating the condition (1) of Definition 1 (groups satisfying this condition are usually said to consist of real elements) and by imposing the condition (2) only on the tensor squares of representations. Clearly, any SR-group is an ASR-group, The converse, generally speaking, is false (see [7]),
The problem posed by J, Saxl is stated as follows: find all finite ASR-groups, We proved the solvability of this class of groups. Note that the class of ASR-groups including also the groups proposed by G.W, Markov.
GG
GG
Furthermore, the theorem allows us to state the following property of irreducible characters of non-solvable groups.
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G
characters x, $ of the group G such that the inner product [x2,$] is greater than 1.
Remark. For infinite groups the assertion of the theorem is false, since the threedimensional rotation group O(3) is a SR-group (see [2]),
§ 3. ASR-groups of odd order
Bv definition, every element of a SR-group is conjugate to its inverse. So any non trivial finite SR-group is of even order. In the definition of ASR-groups this condition is omitted. In [9] it was proved
G
G
characters x, $ of the group G such that the inner product [x2, $] is greater than 1.
There is a following generalization of Theorem 2 Theorem 3 Let G be a finite non-Abelian ASR-group, then |G : G'| = 0 mod 2.
§ 4. Finite supersolvable SR-groups
In this section we state two theorems about the structure of finite supersolvable SR-groups, First of all, we want to recall the following definitions,
• Let p be a prime number, A group G is said to be a p'-group if |G| and p are relatively prime,
• A subgroup H of a group G is called a Hall subgroup if the order of H is coprime to its index,
• The Frattini subgroup $(G) of a group G is the intersection of all its maximal subgroups.
Theorem 4 Let G be a finite supersolvable simply reducible group. Then a 2'-Hall subgroup of G is an Abelian group.
Definition 3 Let Abe an Abelian group and t its involution inverting every element in A. The semidirect product of A and the cyclic group (t) of order 2 is called a generalized dihedral group and is denoted by D(A). [We also admit the possibility |A| = 1. In this case D(A) ~ Z2J.
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Theorem 5 Let G be a finite supersolvable SR-group. Then $(G) is a normal G
G/$(G) - D(Ai) x ... x D(Am),
where m is a non negative integer, D(Aj) are generalized dihedral groups, and Aj are Abelian groups of odd order.
Taking into account Theorems 4 and 5 we see that a finite supersolvable SE-
2
redueible 2-group S there exists a supersolvable SR-group of the form G — D(A) x S, A
5. Some examples of SR- and ASR-groups
A class of finite SR-groups is not empty. Dihedral and generalized dihedral groups, generalized quaternion groups and the symmetric group of degree 4 are simply reducible groups. Direct product of two SR-groups and a factor group of a SR-group are also SR-groups,
Simply reducible 2-groups constitute an important subclass of SR-groups, But the structure of a simply reducible 2-group is not clear at the moment. Using Wigner’s condition and GAP [10] V, V, Yanishevskv found all primary SR-groups of order not greater than 512. In particular he showed that there are 130 SR-groups of order 256, Note that the total number of nonisomorphic groups of order 256 is 56092. The situation with primary ASR-groups is opposite. Almost all 2-groups of a nilpotenev class 2 are ASR-groups. For instance, there are 31742 groups of nilpotenev class 2 of order 256 and only 70 ones are not ASR-groups,
References
1. E, P, Wigner, On representations of certain finite groups, Amer, J, Math, 1941, vol. 63, No. 1, 57-63.
2. M, Hamermesh. Group theory and its application to physical problems Addison-Weslev, Reading, MA-London, 1962,
3. G, W, Markov. Multiplicity free representations of finite groups, Pacific J Math., 1958, vol. 8, 503-510.
4. A. I. Kostrikin. Introduction to algebra. Part III. Basic structures of algebra Fizmatlit, Moscow, 2000. (Russian)
5. V. D. Mazurov and E. I. Khukhro (eds,). Unsolved problems in group theory The Kourovka notebook, 17-th ed,, Institute of Mathematics, Novosibirsk, 2010,
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6. G, W, Mackey. Symmetric and anti-svmmetric Kronecker squares and intertwining numbers of induced representations of finite groups, Amer, J, Math,, 1953, vol. 75, No. 2, 387-405.
7. L. S. Kazarin and V. V. Yanishevskv, On finite simply reducible groups, Algebra i Analiz, 2007, vol. 19, No. 6, 86-116. English transl.: St. Petersburg Math. J., 2008, vol. 19, No. 6, 931-951.
8. L. S. Kazarin, E. I. Chankov, Finite simply reducible groups are solvable, Matem. Sb,, 2010, vol. 201, No. 5, 27-40. English transl.: Sbornik Math., 2010, vol. 201, No. 5, 655-668.
9. L. S. Kazarin, E. I. Chankov. A commutativity criterion for a group of odd order, Modeling and analysis of information systems, 2009, vol. 16, No. 2, 103-108. (Russian)
10. The GAP Group. GAP-Groups, Algorithms and Programming, Version 4.4.12, Aachen, St. Andrews, 2008. URL: http://www.gap-svstem.org
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