What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{|G|}$ holds?
For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds.
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$\begingroup$ As the group ring is identified with the set ${\mathbb K}^{(G)}$ which is the set oif maps from $G$ to $\mathbb K$ with finite support, the answer is: this is the case if and only if the group is finite. $\endgroup$– user1688Commented Jan 25, 2016 at 7:50
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4$\begingroup$ @Anton: if you want an isomorphism as algebras this is not enough. $\endgroup$– Qiaochu YuanCommented Jan 25, 2016 at 7:50
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$\begingroup$ @Qiaochu Yuan: What is the algebra structure on ${\mathbb K}^G$ when $G$ is not finite? $\endgroup$– user1688Commented Jan 25, 2016 at 8:46
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2$\begingroup$ @Anton: pointwise multiplication. $\endgroup$– Qiaochu YuanCommented Jan 25, 2016 at 9:01
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1 Answer
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This is true iff $G$ is finite and abelian, the characteristic of $K$ does not divide $G$, and $K$ has all $n^{th}$ roots of unity whenever $G$ has an element of order $n$. Hopefully it is clear why $G$ must be finite and abelian. The characteristic and root of unity conditions follow from writing $G$ as a product of cyclic groups and applying the Chinese remainder theorem.
answered Jan 25, 2016 at 7:50
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$\begingroup$ Yes sorry I missed about all roots of unity. So is this good enough for algebra isomorphism? $\endgroup$– user76479Commented Jan 25, 2016 at 8:02
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$\begingroup$ @Arul: yes, that's what I'm claiming. $\endgroup$ Commented Jan 25, 2016 at 8:05
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$\begingroup$ Oops yes I didn't say what I meant. I'll rewrite the comment, It' only a minor point anyway, but might as well say it right. $\endgroup$ Commented Jan 25, 2016 at 9:51
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$\begingroup$ (Corrected): I suppose you could simplify the statement a bit: by stating that the field should have n distinct roots of unity whenever $G$ has an element of order $n$, you could omit the statement about the characteristic ( which would be implied by the other conditions). $\endgroup$ Commented Jan 25, 2016 at 9:51
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$\begingroup$ Sure, but might as well make the characteristic hypothesis explicit for readability. $\endgroup$ Commented Jan 25, 2016 at 17:06