Every 1-completed subgroup is contained in a maximal subgroup

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Statement

Every 1-completed subgroup is contained in a maximal subgroup.

In symbols, suppose G is a group and H is a subgroup of G such that there exists x \in G, x \notin H such that \langle H, x \rangle = G. Then, H is contained in a maximal subgroup.

Note that the statement is obviously true for finite groups; the interesting case is that where G is an infinite group (so that it is not necessarily true that every proper subgroup is contained in a maximal subgroup).