Subgroups Of Finite P-groups Research Articles

On intersections of two non-incident subgroups of finite p-groups

Abstract In this paper, we investigate finite p-groups G G such that whenever A , B < G A,B\lt G are non-incident, then A ∩ B ⊴ ⟨ A , B ⟩ A\cap B\hspace{0.33em}⊴\hspace{0.33em}\langle A,B\rangle . This partially solves a problem proposed by Y. Berkovich.

The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups

In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order. However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the Cartesian products of p-groups through their computations. Here, part of our intention is therefore trying to make some designs so as to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. The Cartesian products of p-groups were taken to obtain nilpotent groups. Results up to two dimensions are now obtainable. In this paper, the fuzzy subgroups of the nilpotent product of two abelian subgroups of orders 2^<i>n ^</i>and 128. The integers n ≥ 7 have been successfully considered and the derivation for the explicit formulae for its number distinct fuzzy subgroups were calculated. Some methods were once being used in counting the chains of fuzzy subgroups of an arbitrary finite <i>p</i>-group <I>G. </I>Here, the adoption of the famous Inclusion-Exclusion principle is very necessary and imperative so as to obtain a reasonable, and as much as possible accurate.

The number of conjugacy classes of nonnormal subgroups of finite p-groups (III)

Assume [Formula: see text] is a positive integer and [Formula: see text] is an odd prime. Let [Formula: see text] be the number of conjugacy classes of nonnormal subgroups of a finite group [Formula: see text] and [Formula: see text] is a finite [Formula: see text]-group[Formula: see text]. The set [Formula: see text] has been determined for [Formula: see text]. In this paper, the set [Formula: see text] is determined, and it is discovered that there are three gaps in the values that [Formula: see text] can take in the case of finite [Formula: see text]-groups.

The number of conjugacy classes of nonnormal subgroups of finite p-groups (II)

Assume k is a positive integer and p is an odd prime. Let be the number of conjugacy classes of nonnormal subgroups of a finite group G and is a finite p-group The set have been determined for (see [8]). In this article, the set is determined, and it is discovered that there are two new gaps in the values that can take in the case of finite p-groups.

The lower bound of the number of nonabelian subgroups of finite p-groups

In this paper, we give the lower bound of the number of nonabelian subgroups of possible index of finite p-groups, and classify the finite p-groups such that the number of nonabelian subgroups of possible order are exactly the lower bound.

The intersection of subgroups of finite p-groups

For a finite p-group G and a positive integer k let Ik(G) denote the intersection of all subgroups of G of order pk. This paper classifies the finite p-groups G with \({{I}_k(G)\cong C_{p^{k-1}}}\) for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order pn with \({{I}_k(G)\cong C_{p^{k-\alpha}}}\) are exactly the groups of exponent pn-α.

On subgroups of finite p-groups

In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.

Hall chains in normal subgroups of finite p-groups

Some suffcient conditions for existence of k-admissible Hall chains (= \( \mathcal{H}_k \)-chains) in normal subgroups of finite p-groups are established (for irregular p-groups we consider only the case k = p). In Propositions 13–15 we study p-groups without \( \mathcal{H}_p \)-chains, and metacyclic 2-groups with the above property are classified. Abelian p-groups with exactly one \( \mathcal{H}_k \)-chain are characterized in Proposition 12.

On Subgroups of Finite p-Groups

On Subgroups of Finite p-Groups

Limits of Abelian Subgroups of Finitep-Groups

Abelian subgroups play a key role in the theory and applications of finite p-groups. Our purpose is to establish some very general results motivated by special results that have been of use. In particular, it is w x known KJ that if a finite p-group, for odd p, has an elementary abelian subgroup of order p, n F 5, then it has a normal elementary abelian subgroup of the same order. Our first main result is a general one of this type.

On the poset of conjugacy classes of subgroups of finite p-groups

On the poset of conjugacy classes of subgroups of finite p-groups

Subgroups of finitep-groups inducing the same permutation character

Subgroups of finite_<i>p</i>-groups inducing the same permutation character

Abelian Subgroups of Finite p-Groups

Abelian Subgroups of Finite p-Groups