B.H.Neumann's lemma

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If a group G can be written as a union of left cosets:

G = g_1H_1 \cup g_2H_2 \cup \dots \cup g_nH_n

then G is the union of those g_iH_i for which H_i is a subgroup of finite index in G. In other words, the cosets of subgroups of infinite index are redundant.

Related facts

Facts about unions of subgroups