Abstract Algebra

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Difficult

Abelian Groups Acting Transitively

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Let $A$ be an abelian group of order $a$ acting transitively on a set $X$ of size $n$. Suppose that $A$ acts faithfully (i.e. for every non-trivial $g \in A$ there exists $x \in X$ such that $g \cdot x \neq x$).

Which of the following are possible values for $n$?

A

$a-1$

B

$a$

C

$a(a+1)/2$

D

It depends on whether $A$ is cyclic