The number of homomorphisms from a finite abelian group to a symmetric group

Katsurada, Hidenori; Takegahara, Yugen; Yoshida, Tomoyuki

doi:10.1080/00927870008826958

Cited by 7 publications

(19 citation statements)

References 3 publications

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“…If p = 2, there is a case which does not fall under the conditions of Theorem 2 (more precisely, Condition (2.3) is violated by a small margin), in which one may nevertheless even improve the bound in (2.4). The corresponding theorem, given below, vastly generalises Theorem 1.4 in [10]. We shall apply it in the proof of Theorem 26.…”

Section: P-divisibility Of Coefficients In Exponentials Of Power Serimentioning

confidence: 88%

“…The most general results in this direction, prior to the present paper, are due to Katsurada, Takegahara, and Yoshida, who establish lower bounds for the p-adic valuation v p h n (G) in the case when G is a finite Abelian p-group of rank (minimal number of generators) at most 2; cf. Theorems 1.3 and 1.4 in [10]. For instance, they show that, for a prime p and integers ℓ, m with ℓ ≥ m ≥ 0,…”

Section: Introductionmentioning

confidence: 99%

“…Theorems 2 and 6 in Section 2 vastly generalise these results. We would like to point out though that the proofs of our theorems draw on a key idea in the proof of Theorem 1.2 in [10]: the idea of comparison with a reference power series. In our first main result, Theorem 2, we "truncate" Condition (1.6) in Lemma 1 in the sense that we require (1.6) to hold only up to (and including) the coefficient of z p l −1 , for some given l (see (2.1)), assuming only certain weaker conditions for coefficients of higher powers (see (2.2) and (2.3)).…”

Section: Introductionmentioning

confidence: 99%

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Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

Krattenthaler

Müller

2015

Advances in Mathematics

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Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series S(z) with coefficients in Q p to have coefficients in Z p . We establish theorems on the p-adic valuation of the coefficients of the exponential of S(z), assuming weaker conditions on the coefficients of S(z) than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the p-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian p-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers tion numbers with restrictions on their cycle structure, and a curious "supercongruence" for a certain binomial sum.

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Section: P-divisibility Of Coefficients In Exponentials Of Power Serimentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

Krattenthaler

Müller

2015

Advances in Mathematics

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“…We will give a proof of the fact that ord p (a(n)) ≥ γ(n) (Corollary 3.3 in Section 3). This inequality is shown in [5,6,8,10] and is also a consequence of [7, equation (3)]. If p = 2, this result is equivalent to [3,Theorem 10].…”

Section: Introductionmentioning

confidence: 79%

“…[6,10] Let m be an arbitrary nonnegative integer, and let and u be nonnegative integers less than p. Then…”

Section: So (13) Follows From the Preceding Property Of C(x)mentioning

confidence: 99%

p-Divisibility of the Number of Solutions of xp = 1 in a Symmetric Group

Ishihara¹,

Ochiai²,

Takegahara³

et al. 2001

Annals of Combinatorics

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For a prime p and for the number a(n) of solutions of x p = 1 in the symmetric group on n letters, ord p (a(n)) ≥ [n/p] − [n/p 2 ], and especially, ord p (a(n)) = [n/p] − [n/p 2 ] provided n ≡ 0 mod p 2 . Let r be an integer with 1 ≤ r ≤ p 2 − 1. If ord p (a(r)) ≤ [r/p] + 1, then, for each positive integer m, ord p (a(mp 2 + r)) = m(p − 1) + ord p (a(r)). Assume that ord p (a(r)) = [r/p] + 2. If a(p 2 + r) ≡ −p p−1 a(r) mod p p+[r/p]+2 , then ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2; otherwise, there exists a p-adic integer b such that ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2 + ord p (m − b).

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2-Adic properties for the numbers of representations in the alternating groups

Takegahara

2020

Monatsh Math

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The number of homomorphisms from a finite abelian group to a symmetric group

Cited by 7 publications

References 3 publications

Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

p-Divisibility of the Number of Solutions of xp = 1 in a Symmetric Group

2-Adic properties for the numbers of representations in the alternating groups

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