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Direct product of faithful representations into direct sum 
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#1
Feb1112, 11:27 AM

P: 54

Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct products?
My second question is (if it makes sense), can we have a finite group in which none of the irreducible representations are faithful? Thanks. 


#2
Feb1112, 01:49 PM

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Thus there are lots of other examples, e.g. any noncyclic abelian group, and more generally any group with noncyclic center. 


#3
Feb1112, 04:06 PM

P: 54

Thank you so much! That was totally what I wanted to know.
PS: Yes, I should have written 'tensor product' instead of 'direct product'. 


#4
Feb1412, 05:14 PM

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Direct product of faithful representations into direct sum
No problem. By the way, the end of my first paragraph above should of course read "of a faithful representation V" and not "of a faithful irreducible representation V" (as there might not be such a V! ).



#5
Feb1512, 07:48 AM

P: 54

Yep, understood.
About the cyclic center and having faithful irreducible reps, does this result work if the center is identity? I can find examples of groups in which center is identity; in some cases faithful irreps exist and in some others it doesn't. 


#6
Feb1512, 09:09 AM

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Yes, you're right  the center being cyclic is a necessary but by no means sufficient condition!



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