A finite group $B$ is said to be a B-*group* if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive.

Schur proved that a cyclic group of composite order is a B-group. Wielandt showed that no group of the form $B_1 \times \cdots \times B_d$ with $|B_1| = \ldots = |B_d| \ge 3$ and $d \ge 2$ is a B-group. (See Theorems 25.3 and 25.7 in Wielandt, *Finite permutation groups*.)

Exercise 3.5.6 in Dixon and Mortimer *Permutation groups* asks for a proof that no elementary abelian $p$-group is a B-group. However this is false: the permutation groups of degree $4$ containing $C_2 \times C_2$ as a regular subgroup are $\langle (12)(34), (13)(24)\rangle$ and $\mathrm{Dih}(4)$ (both imprimitive) and $A_4$, $S_4$ (both $2$-transitive). My question is:

For which $d \in \mathbb{N}$ is the elementary abelian group $C_2^d$ a B-group?

Necessary conditions for a primitive permutation groups to have a regular subgroup are given in Corollary 3 of Liebeck, Praeger, Saxl, *Transitive Subgroups of Primitive Permutation Groups*, J. Alg **234**, 291–361 (2000). The main theorem of Li, The finite primitive permutation groups containing an abelian regular subgroup, Proc. London Math. Soc. **87**, 725–747 (2003), gives a sharper result. It seems non-trivial to use either of these papers to answer the question, but I'd welcome a correction.

If $H \le \mathrm{GL}(\mathbb{F}_2^d)$ then the subgroup $\mathbb{F}_2^d \rtimes H$ of the affine general linear group $\mathrm{AGL}(\mathbb{F}_2^d)$ is primitive if and only if $H$ is irreducible. So if there exists an irreducible $H$ acting intransitively on $\mathbb{F}_2^d \backslash \{0\}$, then $C_2^d$ is not a B-group. Such groups exist whenever $d$ is composite, but not always when $d$ is prime. For example, the three irreducible subgroups of $\mathrm{GL}(\mathbb{F}_2^3)$ all contain a Singer element of order $7$, so give $2$-transitive subgroups of $\mathrm{AGL}(\mathbb{F}_2^3)$. (The only other primitive permutation groups of degree $8$ containing a subgroup isomorphic to $C_2 \times C_2 \times C_2$ are $A_8$ and $S_8$, so it follows that $C_2 \times C_2 \times C_2$ is a B-group.) This motivates my second question:

For which primes $p$ is there an irreducible subgroup $H$ of $\mathrm{GL}(\mathbb{F}_2^p)$ such that $H$ acts intransitively on $\mathbb{F}_2^p \backslash \{0\}$?

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