Powers in wreath products of finite groups

1 Introduction

This article deals with computing 𝑟-th powers in the wreath product G ≀ S n , where 𝐺 is a finite group and r ≥ 2 is an integer. In some form or the other, these groups occur as natural subgroups of the symmetric group such as the centralizer of elements, normalizer of certain subgroups, Sylow subgroups and, as such, play an important role in the theory of permutation groups. The group C m ≀ S n is called the generalized symmetric group, where C m is the cyclic group of order 𝑚. Alternatively, C m ≀ S n also appears in the classification of complex reflection groups (see [10, Theorem 8.29]) where it is denoted by G ⁢ ( m , 1 , n ) . In particular, the group C 2 ≀ S n is called the hyper-octahedral group. These groups are also examples of Weyl groups. The Weyl groups of type A n , B n , C n are wreath products, namely S n + 1 , C 2 ≀ S n , C 2 ≀ S n respectively. This investigation of powers in the group G ≀ S n is motivated by the study of powers in S n , in which a substantial amount of information is available. When 𝐺 is trivial, G ≀ S n is S n itself, and therefore our results in this article generalize some of the known results obtained for powers in S n , for example in [4, 6] where the authors have studied the proportion of 𝑟-th powers in S n .

Let r ≥ 2 be any integer. The power map ω r : G ≀ S n → G ≀ S n is defined by ω r ⁢ ( g ) = g r for every g ∈ G ≀ S n . The elements in the image

are called 𝑟-th powers in G ≀ S n . Let P r ⁢ ( G ≀ S n ) := | ω r ⁢ ( G ≀ S n ) | | G | n ⁢ n ! denote the probability that a randomly chosen element of G ≀ S n is an 𝑟-th power. In this article, we give a characterization of 𝑟-th powers in G ≀ S n , assuming 𝑟 is a prime, and thus obtain a generating function for P r ⁢ ( G ≀ S n ) . We use this generating function to prove a partial recursive relation as follows: for a prime r ≥ 2 with ( r , | G | ) = 1 , P r ⁢ ( G ≀ S n ) = P r ⁢ ( G ≀ S n + 1 ) if r ≢ - 1 ⁢ ( mod ⁢ r ) . This is stated as Theorem 5.6. When 𝐺 is trivial, we get back this well-known relation for S n (see [6, Theorem 3.4]).

The set of 𝑟-th powers ω r ⁢ ( G ≀ S n ) is closed under conjugation and therefore is a union of conjugacy classes of G ≀ S n . A conjugacy class of G ≀ S n contained in ω r ⁢ ( G ≀ S n ) is called an 𝑟-th power conjugacy class. A formula is deduced for the number of 𝑟-th power conjugacy classes of G ≀ S n in terms of the function p ⁢ ( n ) , which denotes the number of partitions of a natural number 𝑛. We state and prove this in Theorem 5.9. As stated earlier, the results proved here bring the results obtained for the symmetric group to a more general context. It is worth mentioning that the power map along these lines has been studied for the group GL ⁢ ( n , q ) , which is the group of all n × n invertible matrices over the finite field with 𝑞 elements, by the first author and A. Singh in [9]. In another direction, the authors in [8] showed that asymptotically (more precisely, for large field size), the proportion of powers in finite reductive groups can be essentially computed inside their Weyl groups.

The paper is organized as follows. Section 2 collects some of the important results on powers in the symmetric group. This section serves as a short survey on the topic and also allows us to point out the specific generalizations that we obtain for powers in G ≀ S n . In Section 3, we discuss the notion of cycle index in G ≀ S n , which is the generalization of Pólya’s cycle index for S n . The notion of cycle index will play a major role in this paper. Section 4 is the main technical section where we compute the powers in G ≀ S n and characterize them (see Proposition 4.3). In Section 5, we write a generating function for the proportion of powers in G ≀ S n (see Proposition 5.1) and prove one of the main results, which is Theorem 5.6. A formula for the number of 𝑟-th power conjugacy classes is given in Theorem 5.9. We end this paper with a short discussion on the asymptotics of the powers in G ≀ S n and mention some interesting questions in this direction.

2 Powers in symmetric group

Here, we collect some well-known results on powers in symmetric groups. Let r ≥ 2 be an integer. Consider the power map ω r : S n → S n defined by ω r ⁢ ( π ) = π r for all π ∈ S n . The image of this map, ω r ⁢ ( S n ) = { π r ∣ π ∈ S n } , is the set of 𝑟-th powers in S n . Let P r ⁢ ( S n ) := | ω r ⁢ ( S n ) | n ! be the probability that a randomly chosen element in S n is an 𝑟-th power.

For a natural number 𝑛,

is called a partition of 𝑛 if ∑ i λ i = n . We denote a partition 𝜆 of 𝑛 by λ ⊢ n . The positive integers λ i are called the parts of 𝜆. In power notation, we write

where m i denotes the number of times 𝑖 occur as a part in 𝜆. The conjugacy classes in the group S n are given in terms of the cycle structure of permutations. Any permutation π ∈ S n can be written as a product of disjoint cycles; thus we define type ⁡ ( π ) = ( c 1 , c 2 , … ) , where c i denotes the number of 𝑖-cycles in 𝜋. It satisfies the relation ∑ i i ⁢ c i = n and thus actually defines a partition of 𝑛, that in power notation is 1 c 1 ⁢ 2 c 2 ⁢ … ⊢ n . It is well known that two permutations π , τ ∈ S n are conjugate if and only if type ⁡ ( π ) = type ⁡ ( τ ) . Thus, conjugacy classes in S n are in one-to-one correspondence with partitions of 𝑛.

The 𝑟-th powers in S n can be computed easily if we assume that 𝑟 is a prime. The following lemma, which can be found in [6], will also be useful for our computation in the wreath product. We include a proof since it will be required later.

Using the above lemma, one can easily characterize the permutations which are 𝑟-th powers. The following proposition can be found once again in [6]. We state it without proof.

In [6], the authors showed that the probability of 𝑟-th powers in S n , that is, P r ⁢ ( S n ) , satisfies the following property.

The authors in [6] proved this result using bijective methods. They further investigated P r ⁢ ( S n ) as a sequence in 𝑛 and proved that it is monotonically decreasing in 𝑛 and lim n → ∞ ⁡ P r ⁢ ( S n ) = 0 . In other words, powers in the symmetric groups are really scarce for large 𝑛.

The particular case of r = 2 was investigated by J. Blum in [4], using generating functions. The generating function for the powers in S n can be derived using Polya’s cycle index of S n . We briefly describe the notion of cycle index of S n here (see [16] for more details), for we will see in the next section that this idea can also be generalized for G ≀ S n . The cycle index of S n is defined as

Observe that the product inside is a finite product since there exists m ∈ N such that c i = 0 for all i ≥ m . The coefficient of the monomial

is equal to | Cl ⁡ ( σ ) | n ! , where σ ∈ S n is such that type ⁡ ( σ ) = ( a 1 , a 2 , … ) , and Cl ⁡ ( σ ) denotes the conjugacy class of 𝜎.

The cycle index generating function is given by

Using this, one can write the generating function for the proportion of square permutations (see [4, Equation 5]), which is

Blum used analytic properties of this generating function to give asymptotic estimate of P 2 ⁢ ( S n ) . He proved that

In [15], the author investigated the powers in S n for any r ≥ 2 . He generalized Blum’s estimate for P 2 ⁢ ( S n ) as

where 𝜑 denotes Euler’s phi function and π r is an explicit constant. In [1], the authors have determined the constant π r more explicitly. For more results and estimates on powers in S n and related ideas, we urge the reader to look at [2, 5, 3, 11, 13, 17]. The squares in the alternating group A n are determined in [14].

3 Cycle index of the wreath product G ≀ S n

In this section, we will briefly discuss the notion of cycle index for the wreath product G ≀ S n for a finite group 𝐺, as indicated in [12]. As mentioned before, this is a generalization of the cycle index for S n (see equation (2.1)). Using the cycle index, we will obtain generating functions for powers in G ≀ S n in Section 5.

We start by briefly describing the group G ≀ S n , in the form we will use. We follow the exposition in [7]. Let 𝐺 be a finite group and 𝐻 a permutation group on Ω = { 1 , 2 , … , n } . Define

where f : Ω → G is any function. Define multiplication on G ≀ H as

where f π : Ω → G is defined by f π ⁢ ( i ) = f ⁢ ( π - 1 ⁢ ( i ) ) for every i ∈ Ω and for two functions f , g : Ω → G , f ⁢ g : Ω → G is defined by ( f ⁢ g ) ⁢ ( i ) = f ⁢ ( i ) ⁢ g ⁢ ( i ) . Therefore, f ⁢ f π ′ : Ω → G is defined by f ⁢ f π ′ ⁢ ( i ) = f ⁢ ( i ) ⁢ f ′ ⁢ ( π - 1 ⁢ i ) for every i ∈ Ω . The set G ≀ H with the above binary operation forms a group with ( id , 1 H ) as the identity, where id : Ω → G is defined by id ⁡ ( i ) = 1 G for every i ∈ Ω . For ( f , π ) ∈ G ≀ H , ( f π - 1 - 1 , π - 1 ) is its inverse, where f - 1 : Ω → G is defined by f - 1 ⁢ ( i ) = f ⁢ ( i ) - 1 for every i ∈ Ω . We have f π - 1 - 1 = ( f π - 1 ) - 1 . This completes the description of the group G ≀ H which is called 𝐺 wreath 𝐻. We have | G ≀ H | = | G | n ⁢ | H | . The above description also shows that G ≀ H is the semidirect product of

with 𝐻, under the automorphism ϕ : H → Aut ⁢ ( G n ) , where ϕ h acts on G n by changing indices according to ℎ.

For the rest of the paper, we take H = S n . To define the cycle index of G ≀ S n , we need the description of conjugacy classes in G ≀ S n . We have already seen that conjugacy classes in S n are characterized by the type of a permutation which is just a partition of 𝑛. One can generalize this notion of the type of a permutation to understand the conjugacy classes in G ≀ S n (see [7]). We describe it briefly. Let ( f , π ) ∈ G ≀ S n . Let ( j , π ⁢ ( j ) , … , π t ⁢ ( j ) ) be a cycle appearing in 𝜋 (when written as a product of disjoint cycles). Define the cycle product corresponding to that cycle in 𝜋 as the unique element in 𝐺 given by

Let C 1 , C 2 , … , C s denote a labeling of the conjugacy classes of 𝐺. We define the type of an element ( f , π ) ∈ G ≀ S n as the s × n matrix ( a i ⁢ k ) , where a i ⁢ k is the number of 𝑘-cycles in 𝜋 whose cycle product belongs to C i . If we denote type ⁡ ( π ) = ( c 1 , c 2 , … ) , then it is clearly seen that a i ⁢ k ∈ Z ≥ 0 for every 1 ≤ i ≤ s , 1 ≤ k ≤ n and ∑ i a i ⁢ k = c k for every k = 1 , 2 , … , n . Moreover, ∑ i , k k ⁢ a i ⁢ k = n . The following result that we state without proof determines the conjugacy classes of G ≀ S n and associates combinatorial data to each such conjugacy class.

The type of ( f , π ) is well-defined in the sense that the cycle product does not depend on the representation of a cycle ( j , π ⁢ ( j ) , … , π t ⁢ ( j ) ) in 𝜋. In other words, if ( j , π ⁢ ( j ) , … , π t ⁢ ( j ) ) = ( l , π ⁢ ( l ) , … , π t ⁢ ( l ) ) , then their corresponding cycle products f ⁢ ( j ) ⁢ f π ⁢ ( j ) ⁢ … ⁢ f π t ⁢ ( j ) and f ⁢ ( l ) ⁢ f π ⁢ ( l ) ⁢ … ⁢ f π t ⁢ ( l ) are conjugate in 𝐺. Later, we will see that this in fact gives us a degree of freedom to choose a suitable cycle product in various computations.

Given g = ( f , π ) ∈ G ≀ S n , let Z ⁢ ( g ) denote the centralizer of 𝑔 in G ≀ S n . The structure of the centralizer is well known. For our purpose, we will need the order of the centralizer.

With this information, we can now define the cycle index for G ≀ S n . For convenience, we will denote the type of g ∈ G ≀ S n as T ⁢ ( g ) = ( T ⁢ ( g ) i ⁢ j ) which is an s × n matrix. We define the cycle index of G ≀ S n as

The cycle index of G ≀ S n is a polynomial in the variables t i ⁢ j for 1 ≤ i ≤ s , 1 ≤ j ≤ n . Observe that, given an s × n matrix ( c i ⁢ j ) with ∑ i , j j ⁢ c i ⁢ j = n (that is, it is the type of an element in G ≀ S n ), the coefficient of the monomial ∏ i , j t i ⁢ j c i ⁢ j is equal to | Cl ⁡ ( g ) | | G | n ⁢ n ! , where 𝑔 is such that T ⁢ ( g ) = ( c i ⁢ j ) and Cl ⁡ ( g ) denotes the conjugacy class of 𝑔 in G ≀ S n . Thus, the coefficient of a monomial in Z n is precisely the probability that a randomly chosen element of G ≀ S n belongs to that conjugacy class. The cycle index generating function of G ≀ S n is

We see some examples of cycle index in G ≀ S n .

The following proposition expresses the cycle index generating function of G ≀ S n in an infinite product form, which will be useful later.

Observe that, taking G = { 1 } , the above proposition is the factorization (in infinite product form) for the cycle index generating function of S n (see equation (2.2)).

4 Computing powers in G ≀ S n

In this section, we compute the 𝑟-th powers in G ≀ S n , where r ≥ 2 is prime. We begin with some preparatory lemmas. The proof of the following lemma is easy.

The following lemma is the most important for our purpose. We set an important convention at this point. Given an s × n matrix ( a i ⁢ j ) , we assume a i ⁢ j = 0 for all i > s or j > n . We further assume, for 1 ≤ i ≤ s ,

where C 1 , C 2 , … , C s denote the conjugacy classes of 𝐺. Observe that ( C i ) r = C j for some 1 ≤ j ≤ s .