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Can one give an example of a finite group $G$, with a subset $H$ containing identity, such that $gHg^{-1}=H$ for all $g\in G$, $|H|$ divides $|G|$, but $H$ is not a subgroup of $G$.

Motivation (Theorem of Frobenius): If $G$ acts on a set $X$ tranitively ( $|X|>1$), such that stabilizers are non-trivial but intersection of any two stabilizers is trivial, then the set $K$ of elements of $G$ which have no fixed point, together with identity, form a normal subgroup of $G$. It is easy to see that the condition of normality is easily verified, but to prove that it is a subgroup of $G$, character theory has been used.

While proving this theorem, the necessary conditions are:

  • $|K|$ should divide $|G|$

  • $gKg^{-1}=K$ for all $g\in G$.

I would like to see an example, where subset of $G$ (containing identity) which satisfies these two conditions but it is not a subgroup of $G$.

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Try for Abelian group examples. –  André Nicolas May 28 '11 at 4:55
    
Can someone help improve the title to this question? –  Alexander Gruber Apr 2 '13 at 23:11
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2 Answers

up vote 15 down vote accepted

Consider $G = \displaystyle \frac{\mathbb{Z}}{6\mathbb{Z}} = \bigl\{ \ \overline{0}, \overline{1}, \overline{2}, \cdots, \overline{5}\ \bigr\}$. Consider this subset $H = \bigl\{\ \overline{0},\overline{1} \ \bigr\}$

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Looking for examples in less abelian groups, in $S_5$ we have the the identity, the 24 5-cycles and the 15 3-cycles, giving a subset of size 40. There are examples of size 1680 in $S_7$ and 10080 and 13440 in $S_8$. There is also one of size 112 in ${\rm PGL}(2,7)$. But interestingly I could not find any examples in nonabelian simple groups.

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(Very old thread, but) PSU(3,3) has a normal subset that is half as big as the group: the set of elements of orders 1,2,3,4, or 7 has size 3024 and the group has size 6048. –  Jack Schmidt Jul 9 '13 at 15:51
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