Three subgroups lemma

Notation

• If H and K are subgroups of a group G, the commutator of H and K, denoted by , is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that = 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.
• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by C_G.

  Statement

Proof and the Hall–Witt identity