Timeline for Order of abelian subgroup of the automorphism group of an abelian group
Current License: CC BY-SA 4.0
4 events
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| Mar 14 at 11:45 | comment | added | Sean Eberhard | As for your last question, there are no other examples, but I may offend you by using representation theory again. We know that $B$ acts reguarly on the nonzero points of $V = \mathbb F_p^\ell$, in particular irreducibly, so by Schur's lemma, $K = \mathrm{End}_B(V)$ is a finite division ring, hence a field, and $B \le K^\times$. Since $K^\times$ is abelian and transitive on $V \setminus \{0\}$, $|K^\times| = |V| - 1$. This implies that $B = K^\times$ and $V \cong K$. | |
| Mar 14 at 0:22 | comment | added | tomasz | This is a nice argument, thanks! | |
| Mar 13 at 22:24 | history | edited | KhashF | CC BY-SA 4.0 |
added 6 characters in body
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| Mar 13 at 21:59 | history | answered | KhashF | CC BY-SA 4.0 |