# Special collections of normal subgroups.

Let $G$ be a finite group, and let $N=\{N_1,..., N_n\}$ be a list of nontrivial normal subgroups of $G$ having the following property: For every irreducible representation $\rho$ of $G$ there is some index $j$ such that $\rho$ restricted to $N_j$ is trivial.

Question: Give an example of group $G$ and a collection of nontrivial normal subgroups $N$ of $G$ satisfying the above.

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$Z/2+Z/2$ and all its proper subgroups. – damiano Aug 4 '10 at 17:31
I believe $Z/p+Z/p$, $p$ prime, and all proper subgroups will do. I'm interested to find non-abelian examples. – Justin Greenough Aug 4 '10 at 20:19
Any group wtih non-cyclic center will do. – Steve D Aug 4 '10 at 22:12
Ehehe, it happens! Btw, you might be interested in this thread: mathoverflow.net/questions/29813/… It mentions connected groups, but probably the answers can be, at least partially, adapted. Btw, why do you need such examples? This might guide better the kind of replies you get. – damiano Aug 5 '10 at 12:53
Yes yes of course damiano you are right. I have removed my erroneous comment. Thank you! However I don't know if I understand clearly why having a non-cyclic center implies normal subgroups with the property mentioned in my original question. Such a collection of normal subgroups will provide a covering of the fusion category $Rep(G)$ by fusion (thus braided fusion) subcategories. Really I just need some examples to make an article I'm writing more readable and fun! – Justin Greenough Aug 5 '10 at 13:15

The subgroup generated by the permutations (123), (456) and (23)(56) in $S_6$ has no faithful irreducible representation.