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Weak subgroups

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:

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

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

Sean Eberhard groups, teatime-puzzles Leave a comment 2021-04-262021-04-26

2020 Christmas crossword

Have a go at my Christmas crossword! Get the pdf version, or download the puz file to view with Across Lite or in your browser.

Warning: if you live west of the Atlantic, you may first want to have a look here: https://en.wikipedia.org/wiki/Cryptic_crossword.

Happy holidays!

Sean Eberhard teatime-puzzles Leave a comment 2020-12-252020-12-27

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