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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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    $\begingroup$ $Z/2+Z/2$ and all its proper subgroups. $\endgroup$
    – damiano
    Aug 4, 2010 at 17:31
  • $\begingroup$ I believe $Z/p+Z/p$, $p$ prime, and all proper subgroups will do. I'm interested to find non-abelian examples. $\endgroup$
    – Justin Greenough
    Aug 4, 2010 at 20:19
  • $\begingroup$ Any group wtih non-cyclic center will do. $\endgroup$
    – Steve D
    Aug 4, 2010 at 22:12
  • $\begingroup$ 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. $\endgroup$
    – damiano
    Aug 5, 2010 at 12:53
  • $\begingroup$ 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! $\endgroup$
    – Justin Greenough
    Aug 5, 2010 at 13:15

1 Answer 1

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

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