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High Quality Content by WIKIPEDIA articles! In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall Witt identity. In that which follows, the following notation will be employed: If H and K are subgroups of a group G, the commutator of H and K will be denoted by [H,K]; if L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed. If x and y are elements of a group G, the conjugate of x by y will be denoted by xy. If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).