Finite P-group Research Articles

Rings Whose Invertible Elements Are Weakly Nil-Clean

Westudythoseringsinwhichallinvertible elements are weakly nil-clean, calling them UWNC rings. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as UNC rings. Specifically, our main achievements are that the triangular matrix ring Tn(R) over a ring R is UWNC precisely when R is UNC.Besides, the notions UWNC and UNCdocoincide when2 ∈ J(R). Wealso describe UWNC2-primal rings R by proving that R is a ring with J(R) = Nil(R) such that U(R) = ±1+Nil(R). In particular, the polynomial ring R[x] over some arbitrary variable x is UWNC exactly when R is UWNC. Likewise, we furthermore apply the obtained results to group rings showing that if G is a locally finite p-group and R is a UWNC ring such that the prime p is a nilpotent in R, then RG is too a UWNC ring. Some other relevant assertions are proved in the present direction as well.

ON NONINNER AUTOMORPHISMS OF SOME FINITE P-GROUPS

Abstract We settle the noninner automorphism conjecture for finite p-groups ( $p> 2$ ) with certain conditions. Also, we give an elementary and short proof of the main result of Ghoraishi [‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc.89(2) (2014) 202–209].

On polynomial invariant rings in modular invariant theory

Let k be a field of characteristic p>0, V a finite-dimensional k-vector-space, and G a finite p-group acting k-linearly on V. Let S=SymV⁎. Confirming a conjecture of Shank-Wehlau-Broer, we show that if SG is a direct summand of S, then SG is a polynomial ring, in the following cases:(a)k=Fp and dimk⁡V=4; or(b)|G|=p3. In order to prove the above result, we also show that if dimk⁡VG≥dimk⁡V−2, then the Hilbert ideal hG,S is a complete intersection.

Bounding the exponent of the commutator subgroup of a finite p-group

Bounding the exponent of the commutator subgroup of a finite p-group

The symbol length for elementary type pro-p groups and Massey products

For a prime number p and an integer m≥2, we prove that the symbol length of all elements of m-fold Massey products in H2(G,Fp), for pro-p groups G of elementary type, is bounded by (m2/4)+m. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-p Galois groups G=GF(p) of fields F which contain a root of unity of order p. More generally, we provide such a uniform bound for the symbol length of all pullbacks ρ⁎(ω¯) of a given cohomology element ω¯∈Hn(G¯,Fp), where G¯ is a finite p-group, n≥2, and ρ:G→G¯ is a pro-p group homomorphism.

A class of finite p-groups and the normalized unit groups of group algebras

Let p be a prime and F p be a finite field of p elements. Let F p G denote the group algebra of the finite p-group G over the field F p and V ( F p G ) denote the group of normalized units in F p G . Suppose that G is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p , n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V ( F p G ) p l and G p l , Ω l ( V ( F p G ) ) and Ω l ( G ) are given. Furthermore, there is a direct proof for V ( F p G ) p ∩ G = G p .

Finite p-groups with three character codegrees

For an irreducible character χ of a finite group G, the codegree of χ is defined as | G : ker ( χ ) | / χ ( 1 ) . In this paper, we characterize finite p-groups with three character codegrees.

Finite p-groups satisfying a weak chain condition

Assume G is an A t -group. G satisfies a weak chain condition if its every A i -subgroup is contained in an A i + 1 -subgroup for all i ⩾ 1 . We study two proper subclasses of the class of the A t -groups satisfying a weak chain condition, and the extreme case of the A t -groups satisfying a chain condition.

Criteria for supersolvability of saturated fusion systems

Let p be a prime number. A saturated fusion system F on a finite p-group S is said to be supersolvable if there is a series 1=S0≤S1≤…≤Sm=S of subgroups of S such that Si is strongly F-closed for all 0≤i≤m and such that Si+1/Si is cyclic for all 0≤i<m. We prove some criteria that ensure that a saturated fusion system F on a finite p-group S is supersolvable provided that certain subgroups of S are abelian and weakly F-closed. Our results can be regarded as generalizations of purely group-theoretic results of Asaad [3].

A generalization of Alperin Fusion theorem and its applications

Let F be a saturated fusion system on a finite p-group S, and let P be a strongly F-closed subgroup of S. We define the concept “F-essential subgroups with respect to P” which are some proper subgroups of P satisfying some technical conditions and show that an F-isomorphism between subgroups of P can be factorized by some automorphisms of P and F-essential subgroups with respect to P. When P is taken to be equal to S, the Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P⊴F if and only if there is no F-essential subgroup with respect to P. The following definition is made: A p-group P is strongly resistant in saturated fusion systems if P⊴F whenever there is an over p-group S and a saturated fusion system F on S such that P is strongly F-closed. It is shown that several classes of p-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criterion for a strongly F-closed subgroup to be normal in F. These results are obtained as a consequence of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting in its own right.

Automorphic word maps and the Amit–Ashurst conjecture

Abstract In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.

On commuting automorphisms of finite p-groups of coclass 3

Let [Formula: see text] be a group. If the set [Formula: see text] forms a subgroup of [Formula: see text], then G is called [Formula: see text]-group. Recently, it has been proven that the minimum coclass of a non-[Formula: see text][Formula: see text]-group is equal to [Formula: see text]. In this paper, we deal with finite [Formula: see text]-groups of coclass [Formula: see text]. We prove that a finite [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text], of coclass [Formula: see text] for an odd prime [Formula: see text] is an [Formula: see text]-group except when the nilpotency class of the second term of upper central series [Formula: see text], [Formula: see text], is equal to [Formula: see text]. In particular, we show that in this case, center of [Formula: see text], [Formula: see text], and commutator subgroup [Formula: see text], [Formula: see text], are of order [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] is a group of order [Formula: see text]. Also, we give some sufficient conditions on a finite 2-group [Formula: see text] of coclass [Formula: see text] implying that [Formula: see text] is an [Formula: see text]-group.

On the isomorphism problem for certain p-groups

We consider 9 infinite families of finite p-groups, for p a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified.

On the Hughes conjecture for some finite p-groups

On the Hughes conjecture for some finite p-groups

Control of transfer for odd primes

Suppose p is a prime and T is a non-identity finite p-group. We show that if p is odd, then there exist two non-identity characteristic subgroups K1(T) and K2(T) of T such that K1 and K2 jointly control transfer in every finite group G containing T as a Sylow p-subgroup, in the sense that the normalizers of K1(T) and K2(T) in G determine the largest abelian factor group of G that is a p-group. We also give a new example of a characteristic subgroup K(T) of T such that K controls transfer in G by itself if p⩾5.

SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS

Abstract Suppose that G is a finite solvable group. Let $t=n_c(G)$ denote the number of orders of nonnormal subgroups of G. We bound the derived length $dl(G)$ in terms of $n_c(G)$ . If G is a finite p-group, we show that $|G'|\leq p^{2t+1}$ and $dl(G)\leq \lceil \log _2(2t+3)\rceil $ . If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of $|G'|$ is less than t and that $dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$ .

The Modular Isomorphism Problem and Abelian Direct Factors

Let p be a prime and let G be a finite p-group. We show that the isomorphism type of the maximal abelian direct factor of G, as well as the isomorphism type of the group algebra over Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p$$\\end{document} of the non-abelian remaining direct factor, if existing, are determined by FpG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p G$$\\end{document}, generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra 636, 533-559 (2023)) over the prime field. To do this, we address the problem of finding characteristic subgroups of G such that their relative augmentation ideals depend only on the k-algebra structure of kG, where k is any field of characteristic p, and relate it to the modular isomorphism problem, extending and reproving some known results.

Abelian invariants and a reduction theorem for the modular isomorphism problem

We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite p-group. Finally, we apply our results to new classes of groups.

Finite 𝑝-groups of class two with a small multiple holomorph

Abstract We consider the quotient group T ⁢ ( G ) T(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T ⁢ ( G ) T(G) contains a cyclic subgroup of order p r − 1 ⁢ ( p − 1 ) p^{r-1}(p-1) , where p r p^{r} is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with r = 1 r=1 ) such that T ⁢ ( G ) T(G) has order exactly p − 1 p-1 , which is as small as possible.

The socle of the group algebra of a finite p-group

Let G be a finite p-group, and α an automorphism of the group algebra FpG. Then α fixes the socle of FpG pointwise. More generally, if k is a field of characteristic p, and α is a k-algebra automorphism of kG, then α induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the (p−1)st power of the product of the determinants of this action. The scalar is thus an element of (k×)p−1.