Weak subgroups

groups, teatime-puzzles

Here is a teatime puzzle.

Suppose {G} is a finite group and {S \subseteq G} is a subset containing the identity such that {xy \in S} whenever {x \in S}, {y \in S}, and {x \neq y}. Show that these are the only possibilities:

  1. {S} is a subgroup,
  2. {S = \{1, x\}} for some {x \in G},
  3. {S = \{1, x, x^{-1}\}} for some {x \in G},
  4. {S = Q \setminus \{-1\}} for some quaternion subgroup {Q \leq G}.

What other possibilities are there if we do not assume {1 \in S}?

Published by Sean Eberhard

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