Narrow Results

A group G is said to have finite section p-rankr_p(G) = r (here p is a prime) if every elementary abelian p-section U/V of G is finite of order at most p^r and there is an elementary abelian p-section A/B of G such that |A/B| = p^r. If ℙ is the set of all primes and λ : ℙ → ℕ ∪ {0} is a function, we say that a group G has λ-bounded section rank if r_p(G) ≤ λ(p) for each p ∈ ℙ. In this paper we show that if G is a locally generalized radical group in which the normal closures of the cyclic subgroups of G have finite λ-bounded section rank, then [G,G] has -bounded section rank for some function . This is a wide generalization of some results by Neumann, Smith and many others. Moreover we are able to give explicit formulas for the involved bounds.

It had been proved by Birman and Goldberg that the normal closure of the pure braid group $P_n(D)$ in the pure braid group of the torus $P_n(T)$ is the commutator subgroup $[P_n(T),P_n(T)]$. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of $B_n(D)$ in $B_n(T)$, which turns out to have an interesting geometric description.

We describe the structure of finite groups in which all normal closures of non-normal subgroups have the same order. In particular, in the case of $p$-groups, we give a complete classification.

A gyrogroup is a non-associative structure that may be regarded as a suitable generalization of groups. In this paper, we introduce the notion of an associator in an arbitrary gyrogroup that, in some sense, measures the deviation from associativity of gyrogroup operations. We then study the normal closure of the set of associators and establish several relevant properties, including the universal property of the associativization of a gyrogroup. This leads to a few invariant properties of gyrogroups in terms of associator normal subgyrogroups and gives an effective method to study representations of gyrogroups.