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Let $G$ be a finite group and $\mathbb{F}_q$ be a finite field, where $q$ is a prime power. Let $\mathbb{F}_qG$ be a finite semisimple group algebra.

Then Wedderburn decomposition theorem implies that $\mathbb{F}_qG$ is direct sum of matrix rings over division rings, i.e.,$$\mathbb{F}_qG\cong \oplus_{i=1}^t M(n_i, \mathbb{F}_{q_i}).$$

Here $\mathbb{F}_{q_i}$ is a finite extension of $\mathbb{F}_{q}$, $n_i\geq 1$ and $t\geq 1$ are positive integers.

Given: Suppose that $\mathbb{F}_{q}$ is such that $$\mathbb{F}_qG\cong \oplus_{i=1}^t M(n_i, \mathbb{F}_{q}).$$ This means $q_i=q$ for each $i$.

Let $q'\neq q$ be another prime power such that the group algebra $\mathbb{F}_{q'}G$ is again semisimple.

Suppose that $M\left(n, \mathbb{F}_{(q')^r}\right)$ is a Wedderburn component of the group algebra $\mathbb{F}_{q'}G$ for some $r\geq 2$ and any positive integer $n$.

Then I want to show that $M\left(n, \mathbb{F}_{q}\right)$ must be a Wedderburn component of the group algebra $\mathbb{F}_qG$.

I have tried various examples and found this result true. Please help me to prove this.

TShiong
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PAMG
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1 Answers1

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First of all, I just reminded you that $\mathbb{F}_qG$ is semisimple if and only if $p$ and $|G|$ are co-prime.

Now for the question. The idea is to tensor with $\overline{\mathbb{F}_q}$. It's easy to see that $$\overline{\mathbb{F}_q}G =\mathbb{F}_qG\otimes_{\mathbb{F}_q}\overline{\mathbb{F}_q}\cong \bigoplus M(n_i,\overline{\mathbb{F}_q})$$ On the other hand, we get $$\overline{\mathbb{F}_{q}}G =\mathbb{F}_{q'}G\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q}\cong \bigoplus M(m_i,\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q})$$ Now using that $\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q}\cong \overline{\mathbb{F}_q}^{r_i} $, we can get $$M(m_i,\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q})\cong M(m_i,\overline{\mathbb{F}_q})^{r_i}$$ From the uniqueness of the decomposition, we get that every $m_i$ is a some $n_i$.

For the claim about the tensor product of the fields look at étale algebres or this post.

re'em waxman
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  • Can you please tell some references where I can get to study the properties of tensor products mentioned by you? – PAMG Apr 21 '23 at 01:18
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    Most of the properties are standards like $(A\oplus B)\otimes C = (A\otimes C)\oplus (B\otimes C)$. Every algebra book introducing tensor products (like algebra by Lang) will prove them. I also used that $M(n,A)\otimes_{B}C\cong M(n, A\otimes_{B}C)$, it's a nice exercise proving this. – re'em waxman Apr 21 '23 at 10:55