I'm trying to find an example of a group $G$ such that $|G| = 120$, and a non-normal subgroup $H$ within it. Of course, my first instinct is to let $G = S_5$, but this doesn't work because all subgroups of $S_5$ are normal. Help would be greatly appreciated!
Example of a Group with Certain Characteristics
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group-theory
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0Conjugating rho=(12345) will preserve its cycle type, but the result will not necessarily be a power of (12345). (Do some computations for yourself to check.) As it is a 5-cycle, the subgroup generated by it will have 5 elements, whereas the conjugacy class associated to it has 5!/5=4!=24 members. As Don noted, conjugacy classes are not the same thing as subgroups, though conjugacy classes of the symmetric groups are always those permutations of a given cycle type as you observe. – 2012-12-16
1 Answers
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$S_3$ has $6$ elements and a few non-normal subgroups.
Let $H$ be one of the subgroups that is not normal in $S_3$.
Consider next the group $G := S_3 \times \mathbb{Z}_{20}$, which has $120$ elements.
Then $H \times\{0\}$ is a non-normal subgroup of $G$.