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I am looking for examples of socle and normal-subgroup relations.

If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) = Soc(A_{4}) = V_{4}$.

Now I am looking for some easy example where this does not work. Where we have no normal subgroup in $G$, s.t. the cut with the $Soc(G)$ equals the socle of this normal subgroup.

Any hints where I can look for such a group?

Thanks and best :) Kathrin

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    Ah ok, so for $N = $ for example, the minimal normal subgroups are $$ and $$ s.t $Soc(N) = \{e, S, T^{2}, ST^{2}\}$? Thanks a lot for all the help :)2012-11-21

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