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Commuting elements with respect to the operator $\wedge $ in infinite groups
Rashid Rezaei,Francesco G. Russo 대한수학회 2016 대한수학회보 Vol.53 No.5
Using the notion of complete nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G$ ($p$ prime), we develop the theory of the exterior degree $\widehat{\mathrm{d}}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs $\mathrm{d}(G)$. Among the main results of this paper, we describe upper and lower bounds for $\widehat{\mathrm{d}}(G)$ with respect to $\mathrm{d}(G)$. Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the $p$-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on $G$ and $\widehat{\mathrm{d}}(G)$.
Commuting powers and exterior degree of finite groups
Peyman Niroomand,Rashid Rezaei,Francesco G. Russo 대한수학회 2012 대한수학회지 Vol.49 No.4
Recently, we have introduced a group invariant, which is re-lated to the number of elements x and y of a nite group G such that x ^ y = 1G^G in the exterior square G ^ G of G. This number gives re-strictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form hm ^ k of H ^ K such that hm ^ k = 1H^K, where m 1 and H and K are arbitrary subgroups of G.
COMMUTING ELEMENTS WITH RESPECT TO THE OPERATOR Λ IN INFINITE GROUPS
Rezaei, Rashid,Russo, Francesco G. Korean Mathematical Society 2016 대한수학회보 Vol.53 No.5
Using the notion of complete nonabelian exterior square $G\hat{\wedge}G$ of a pro-p-group G (p prime), we develop the theory of the exterior degree $\hat{d}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for $\hat{d}(G)$ with respect to d(G). Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the p-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on G and $\hat{d}(G)$.
On the topology of the nonabelian tensor product of profinite groups
Francesco G. Russo 대한수학회 2016 대한수학회보 Vol.53 No.3
The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-$p$-group, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.
ON THE TOPOLOGY OF THE NONABELIAN TENSOR PRODUCT OF PROFINITE GROUPS
Russo, Francesco G. Korean Mathematical Society 2016 대한수학회보 Vol.53 No.3
The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-p-group, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.
COMMUTING POWERS AND EXTERIOR DEGREE OF FINITE GROUPS
Niroomand, Peyman,Rezaei, Rashid,Russo, Francesco G. Korean Mathematical Society 2012 대한수학회지 Vol.49 No.4
Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.
Some algebraic and topological properties of the nonabelian tensor product
Daniele Ettore Otera,Francesco G. Russo,Corrado Tanasi 대한수학회 2013 대한수학회보 Vol.50 No.4
Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.
SOME ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF THE NONABELIAN TENSOR PRODUCT
Otera, Daniele Ettore,Russo, Francesco G.,Tanasi, Corrado Korean Mathematical Society 2013 대한수학회보 Vol.50 No.4
Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.