Groups of prime-power order with a small second derived quotient
نویسنده
چکیده
For odd primes we prove some structure theorems for finite p-groupsG, such that G′′ 6= 1 and |G′/G′′| = p. Building on results of Blackburn and Hall, it is shown that γ3(G) is a maximal subgroup of G ′, the group G has a central decomposition into two simpler subgroups, and, moreover, G′ has one of two isomorphism types.
منابع مشابه
Finite groups with $X$-quasipermutable subgroups of prime power order
Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...
متن کاملA Characterization of the Small Suzuki Groups by the Number of the Same Element Order
Suppose that is a finite group. Then the set of all prime divisors of is denoted by and the set of element orders of is denoted by . Suppose that . Then the number of elements of order in is denoted by and the sizes of the set of elements with the same order is denoted by ; that is, . In this paper, we prove that if is a group such that , where , then . Here denotes the family of Suzuk...
متن کاملA note on power values of generalized derivation in prime ring and noncommutative Banach algebras
Let $R$ be a prime ring with extended centroid $C$, $H$ a generalized derivation of $R$ and $ngeq 1$ a fixed integer. In this paper we study the situations: (1) If $(H(xy))^n =(H(x))^n(H(y))^n$ for all $x,yin R$; (2) obtain some related result in case $R$ is a noncommutative Banach algebra and $H$ is continuous or spectrally bounded.
متن کاملOn counting and generating curves over small finite fields
We consider curves defined over small finite fields with points of large prime order over an extension field. Such curves are often referred to as Koblitz curves and are of considerable cryptographic interest. An interesting question is whether such curves are easy to construct as the target point order grows asymptotically. We show that under certain number theoretic conjecture, if q is a prim...
متن کاملOn the Concept of Co-Sets in a Semi-Group.
and a group of type ln it cannot have the property that each of its subgroups of composite order is invariant. If the order of G is not a power of a prime number each of its Sylow subgroups whose order exceeds a prime number is invariant under G and hence G involves invariantly the direct product of all its Sylow subgroups of composite orders. The quotient group of G with respect to this direct...
متن کامل