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2 Reformulated the question

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups.

Let ${p}$ be a prime, and let $\mathbb{F}_p$ be the field of ${p}$ elements. Let $G,H$ be finite groups, $p$-groups, and let $\mathbb{k}[G]$ denote the group algebra.

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

If not, what if $G$ is a (finite) $p$-group?

1

# Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Let ${p}$ be a prime, and let $\mathbb{F}_p$ be the field of ${p}$ elements. Let $G,H$ be finite groups, and let $\mathbb{k}[G]$ denote the group algebra.

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

If not, what if $G$ is a (finite) $p$-group?