# How to express elements in the intersection of two normal subgroups

If group G has a set of generators and set of relations, and given H, K two normal subgroups. Suppose one can write down the elements in H and K explicitly (also in terms of generators and relations). Now how to obtain an element which is in the intersection H\cap K but not in the commutator subgroup [H,K]?

• Your question is very vaguely stated. An element of the intersection is precisely an element of the kernel of the product map $G\to G/H\times G/K$. Does that help? – HJRW Apr 3 '12 at 20:26
• Why do you think there is such element? A priori $H \cap K=[H,K]$, e.g. $H=K=G$ and $G$ is perfect. So there cannot be a general procedure without more conditions. – Yiftach Barnea Apr 18 '12 at 7:34

R. Brown Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', Topology 23 (1984) 337-345.
Denote the classifying space of a group $G$ by $BG$. Given normal subgroups $M,N$ of $G$ one can form the space $X$ as the homotopy pushout (i.e. double mapping cylinder) of the two maps $BG \to B(G/M), BG \to B(G/N)$. Then the second homotopy group of $X$ is isomorphic to
$$(M \cap N)/ [M,N] .$$
Actually the result of the paper says more, namely that the homotopy 2-type of $X$ is described by the crossed module $M \circ N \to G$ where $\circ$ is the coproduct of the title of the paper. It is feasible that the question asked could be helped by an analysis of this coproduct. (Note that normal subgroups of a group are special cases of crossed modules.)
• Thank you! Indeed I was considering the condition that $(M\cap N)=[M,N]$. I will check your paper. – Changlong Apr 4 '12 at 16:18