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A p-group is a group of order $p^d$ where p is a prime.

If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible representation of the center which would map a generator of the center to $e^{2\pi i/m}$. Could we then induce a representation on the whole group? If so, how do we know this is faithful and irreducible? If not, how else could we prove this?

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    $\begingroup$ Inducing characters from the center behaves fairly nicely (it just gives you a multiple of the original character), and inducing a faithful character always gives a faithful character. $\endgroup$ – Tobias Kildetoft May 29 '13 at 14:13
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The induced representation is faithful, but not necessarily irreducible. But, since the centre of the $p$-group $P$ is cyclic, the unique subgroup $K$ of $Z(P)$ with $|K|=p$ is contained in every nontrivial normal subgroup of $P$. Since the induced representation is faithful, at least one of its irreducible constituents does not have $K$ in its kernel, and then that consituent must be faithful.

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  • $\begingroup$ Will the induced character not just be a multiple of some irreducible character (in which case the faithfulness of that irreducible character is clear)? $\endgroup$ – Tobias Kildetoft May 29 '13 at 14:26
  • $\begingroup$ Yes you are right! $\endgroup$ – Derek Holt May 29 '13 at 18:47
  • $\begingroup$ Can I ask, what is the induced representation given by? I don't see this. $\endgroup$ – user117449 Mar 17 '16 at 20:02
  • $\begingroup$ IThe centre $Z(P)$ is cyclic and has a faithful representation $\rho$. Then we take the induced representation $\rho_H^G$, which is a faithful representation of $G$. You will need to learn about induced representationsa if you do not know what they are. $\endgroup$ – Derek Holt Mar 17 '16 at 20:22
  • $\begingroup$ Thank you. Where can I read about this? Can you explain briefly how the induced representation is constructed? $\endgroup$ – user117449 Mar 17 '16 at 20:37
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T(A)T(B)=T(AB) If all the matrics are distinct than T is called group and there are one to one correspondance between matrix's and group element of G . So there is a isomerism bet. G and T .if isomerism is perfect T and G thn this type of representation is called faithfully representation of group G.

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