I am reading James and Liebeck's book on Representations and Characters of Groups.(adsbygoogle = window.adsbygoogle || []).push({});

Exercise 1 of Chapter 3 reads as follows:

Let G be the cyclic group of order m, say G = < a : [itex] a^m = 1 [/itex] >.

Suppose that A [itex] \in GL(n \mathbb{C} ) [/itex], and define [itex] \rho : G \rightarrow GL(n \mathbb{C} ) [/itex] by

[itex] \rho : a^r \rightarrow A^r \ \ (0 \leq r \leq m-1 ) [/itex]

Show that [itex] \rho [/itex] is a representation of G over [itex] \mathbb{C} [/itex] iff and only if [itex] A^m = I [/itex]

The solution given is as follows (functions are applied from the right)

Suppose [itex] \rho [/itex] is a representation of G. Then

[itex] I = 1 \rho = ( a^m ) \rho = {(a \rho)}^m = A^m [/itex]

Conversely assume that [itex] A^m = I [/itex]. Then [itex] ( a^i ) \rho = A^i [/itex] for all integers i.

Therefore for all integers i, j

[itex] ( a^i a^j ) \rho \ = \ ( a^{i+j} ) \rho \ = \ A^{i+j} \ = \ A^i A^j \ = \ ( a^i \rho ) a^j \rho ) [/itex]

and so [itex] \rho [/itex] is a representation.

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That is all fine but I am just getting into representations and wish to get an intuitive understanding of what is happening.

In the above I am suprised that we have no explicit form for A.

Does this mean we have many representations for G in this case - that is any and every matrix A for which [itex] A^m = I [/itex]

Can someone please confirm this?

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# Representations of the cyclic group of order n

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