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In this paper, we introduce a condition on a p-subgroup of a finite group that is more general than the concept of ICΦ-subgroups, and obtain a sufficient and necessary condition for p-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a finite group G is isomorphic to the alternating group An with n≥5 if and only if they have the same set of the orders of solvable subgroups. [ABSTRACT FROM AUTHOR]

A binomial f belonging to a toric ideal I is indispensable if, for any system $\mathcal{F}$ of binomial generators of I, either f or -f belongs to $\mathcal{F}$. In the present paper, we study indispensable binomials of the toric ideals IG arising from a finite graph G. First, we show that the toric ideal IG arising from a finite graph G whose complementary graph is weakly chordal is generated by the indispensable binomials if and only if no complete graph of order ≥4 is a subgraph of G. Second, we completely classify indispensable binomials of the toric ideal IG arising from a finite graph G satisfying the odd cycle condition. Finally, the existence of indispensable binomials of IG will be discussed. [ABSTRACT FROM AUTHOR]

A subnormal subgroup X of G has the following property: there exists a Dirichlet polynomial Q(s) with integer coefficients such that, for each t ∈ ℕ, Q(t) is the conditional probability that t random elements generate G given that they generate G together with the elements of X In this paper we analyze how far can a subgroup X be with this property from being a subnormal subgroup. [ABSTRACT FROM AUTHOR]

Let Γ be a totally ordered abelian group of finite rational rank r. Consider s and n two non-negative integers with r+s≤n and denote by K a field. The main purpose of this paper is to construct a s-dimensional valuation v of the quotient field K((X1,...,Xn)) of the corresponding power series ring K[[X1,...,Xn]] such that v birrationally dominates K[[X1,...,Xn]] and has Γ as group of values. This is possible except for the case in which Γ=ℤ is the group of the integers, s=0, n≥2 and K does not admit an infinite algebraic extension. In this last case, we show that there does not exist such a valuation. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co2 and Co3. Computations were carried with the aid of computer algebra system 픾픸ℙ [18]. [ABSTRACT FROM AUTHOR]