Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

In this article we introduce the notion of n-capable groups. It is shown that every group G admits a uniquely determined subgroup (〖Z^n)〗^* (G) which is a characteristic subgroup and lies in the n-centre subgroup of the group G. This is the smallest subgroup of G whose factor group is n-capable. Moreover, some properties of n-central extension will be studied.

Let p be a prime number and F an algebraic closure of the finite field Fp with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group GQ = Gal(Q̄/Q) into GL(n,F), unramified at all finite primes not dividing pN . (We shall say “ρ is unramified outside pN .”) In this paper, “representation” will always mean “continuous, semisimple ...

A finite transitive permutation group is said to be elusive if it has no fixed point free elements of prime order. In this paper we show that all elusive groups G = N oG1 with N an elementary abelian minimal normal subgroup and G1 cyclic, can be constructed from transitive subgroups of AGL(1, p2), for p a Mersenne prime, acting on the set of p(p + 1) lines of the affine plane AG(2, p).

In two previous papers we computed cohomology groups H(Γ0(N);C) for a range of levels N , where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N . In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the pap...

/ = /i$i. Since (to)i = 5ii is in (s), and ni is prime to the order of Si, Si is a power of s. Thus h, as well as t, corresponds to t in the isomorphism of G with G/(s) ; but h and /' are of the same order tii. Every element of G/(s) whose order is a divisor of m/mi corresponds to an element of G whose order is a divisor of m. It follows that t', and hence every element of G/(s) whose order div...

a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

We give an estimate for the width of congruence subgroup SL(n,OS,I) in Tits–Vaserstein generators, where OS is a localization ring integers number field K. assume that either K has real embedding, or ideal I prime to roots unity