arXiv subjects
Publications and source records attributed to Yassine Guerboussa.
We show that every finitely generated cohomologically trivial module over $RG$, where $G$ is a finite $p$-group and $R$ is a $p$-adic ring, splits as the direct sum of a finite cohomologically trivial $RG$-module and a free $RG$-module. Along the way, we also establish other results concerning generators and relators of such modules.
We classify, up to isomorphism, the $\mathbb{Z}_pG$-modules of rank $1$ (i.e., the quotients of $\mathbb{Z}_pG$) for $G$ cyclic of order $p$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. This allows us in particular to determine effectively the quotients of $\mathbb{Z}_pG$ which are cohomologically trivial over $G$. There are natural zeta functions associated to $\mathbb{Z}_pG$ for which we give an explicit formula.
A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields the natural problem of comparing the invariants of $A$ as a $\mathbb{Z}_p$-module to its invariants as an $\mathcal{O}$-module. We solve the latter problem in a more general context, and give some applications to the structure of some $p$-groups and their automorphisms.
A finite $p$-group $G$ is said to be $d$-maximal if $d(H)<d(G)$ for every subgroup $H<G$, where $d(G)$ denotes the minimal number of generators of $G$. A similar definition can be formulated when $G$ is acted on by some group $A$. We generalize results of B. Kahn and T. Laffey to the latter case, and give them in particular alternative short proofs. We answer moreover a question of Y. Berkovich about the minimal non-metacyclic $p$-groups.
We prove a cohomological property for a class of finite $p$-groups introduced earlier by M. Y. Xu, which we call semi-abelian $p$-groups. This result implies that a semi-abelian $p$-group has non-inner automorphisms of order $p$, which settles a longstanding problem for this class. We answer also, independetly, an old question of M. Y. Xu about the power structure of semi-abelian $p$-groups.
Let $G$ be a finite $p$-group acted on faithfully by a group $A$. We prove that if $A$ fixes every element of order dividing $p$ ($4$ if $p=2$) in a specified subgroup of $G$, then both $A$ and $[G,A]$ behave regularly, that is the elements of order dividing any power $p^i$ in each one of them form a subgroup; moreover $A$ and $[G,A]$ have the same exponent, and they are nilpotent of class bounded in terms of $p$ and the exponent of $A$. This leads in particular to a solution of a problems posed by Y. Berkovich. In another direction we discuss some aspects of the influence of a $p$-group $P$ on the structure of a finite group which contains $P$ as a Sylow subgroup, under assumptions like every element of order $p$ ($4$ if $p=2$) in a given term of the lower central series of $P$ lies in the center of $P$.
We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of $p$-groups. These facts and some of their applications are investigated in this paper.
Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ σ\mapsto h_σ$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_σ(x)=x^{-1} σ(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the main tool in our approach.
A conjecture of Berkovich asserts that every non-simple finite p-group has a non-inner automorphism of order p. This conjecture is far from being proved despite the great effort devoted to it. In this paper we prove it for p-groups of coclass 2, provided that p is odd. Some related results are also proved, and may be considered as interesting independently.