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I am looking for a non-semisimple finite-dimensional complex representation of a locally finite group. A concrete example would be great, but I would also appreciate references.

I know that the group must be uncountable, since finite-dimensional complex representations of countable locally finite groups are all semisimple.

My motivation for seeking these examples comes from the article "Representations of locally finite groups" by Winter, which shows that any finite-dimensional complex semisimple representation of a locally finite group is an inflation from a countable quotient. Hence I wondered about the non-semisimple ones.

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If $G$ is a locally finite group then every finite dimensional complex representation of $G$ is semisimple.

Suppose $U\leq V$ are finite dimensional complex representations of $G$. For each finite subset $X$ of $G$, let $S_X$ be the subset of $\operatorname{Hom}_{\mathbb{C}}(V,U)$ consisting of maps that split the inclusion map and which commute with all elements of $X$. Then $S_X$ is nonempty, by Maschke’s Theorem for the finite group generated by $X$, and is an affine subspace of $\operatorname{Hom}_{\mathbb{C}}(V,U)$.

Choose $X$ so that the dimension of $S_X$ is minimal. Then for $X\leq Y$ we must have $S_X=S_Y$. So any element of $S_X$ commutes with all elements of $G$, and so $U$ is a direct summand of $V$ as representations of $G$.

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