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Let $G$ be the finite simple group of Lie type $G = E_7(q)$, where $q$ is an odd prime power. Then $G$ is an index $2$ subgroup of the adjoint group $G_{\operatorname{ad}}$, which is also denoted by $G_{\operatorname{ad}} = \operatorname{Inndiag}(G)$ and known as the group of inner-diagonal automorphisms. It was proven by Cohen--Liebeck--Saxl--Seitz (1992) that there is an elementary abelian $2$-subgroup $E$ of order $4$ in $G_{\operatorname{ad}}$, such that $N_{G_{\operatorname{ad}}}(E)/C_{G_{ad}}(E) \cong \operatorname{Sym}_3$, and $C_{G_{\operatorname{ad}}}(E) = E \times \operatorname{Inndiag}(D_4(q))$. Furthermore, such an $E$ is unique up to conjugacy in $G_{\operatorname{ad}}$. It is known that $N_G(E)$ is always a maximal subgroup of $G$, and $N_{G_{\operatorname{ad}}}(E)$ is a maximal subgroup of $G_{\operatorname{ad}}$ unless $N_{G_{\operatorname{ad}}}(E) \leq G$. In this note, we describe the structure of $N_{G}(E)$. It turns out that $N_G(E) = N_{G_{\operatorname{ad}}}(E)$ if and only if $q \equiv \pm 1 \mod{8}$., Comment: 9 pages

We continue classification of finite groups which can be used as symmetry group of the scalar sector of the four-Higgs-doublet model (4HDM). Our objective is to systematically construct non-abelian groups via the group extension procedure, starting from the abelian groups $A$ and their automorphism groups $\mathrm{Aut}(A)$. Previously, we considered all cyclic groups $A$ available for the 4HDM scalar sector. Here, we further develop the method and apply it to extensions by the remaining rephasing groups $A$, namely $A = \mathbb{Z_2}\times\mathbb{Z_2}$, $\mathbb{Z_4}\times \mathbb{Z_2}$, and $\mathbb{Z_2}\times \mathbb{Z_2}\times \mathbb{Z_2}$. As $\mathrm{Aut}(A)$ grows, the procedure becomes more laborious, but we prove an isomorphism theorem which helps classify all the options. We also comment on what remains to be done to complete the classification of all finite non-abelian groups realizable in the 4HDM scalar sector without accidental continuous symmetries., Comment: 27 pages, 0 figures, 4 tables. v2: extra clarifications, matches the published version

We prove that if $G$ is a finite irreducible solvable subgroup of an orthogonal group $O(V,Q)$ with $\dim V$ odd, then $G$ preserves an orthogonal decomposition of $V$ into $1$-spaces. In particular $G$ is monomial. This generalizes a theorem of Rod Gow., Comment: to appear in Bull. Lond. Math. Soc

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$. For unipotent elements $u \in G$ and nilpotent elements $e \in \mathfrak{g}$, the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ are known in most cases. In the cases that remain, the group $G$ is of classical type in bad characteristic, so $\operatorname{char} K = 2$ and $G$ is of type $B_{\ell}$, $C_{\ell}$, or $D_{\ell}$. In this paper, we consider the case where $G$ is of type $C_{\ell}$ and $\operatorname{char} K = 2$. As our main result, we determine the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ for all unipotent $u \in G$ and nilpotent $e \in \mathfrak{g}$. In the case where $G$ is of adjoint type, we will also describe the Jordan block sizes on $[\mathfrak{g}, \mathfrak{g}]$., Comment: 39 pages

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., Comment: 26 pages

Let $G$ be an irreducible imprimitive subgroup of $\operatorname{GL}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that $G$ can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where $G$ is the wreath product of an irreducible primitive $H \leq \operatorname{GL}_d(\mathbb{F})$ and transitive $K \leq S_k$, where $n = dk$. We show that $G$ has a unique nonrefinable system of imprimitivity, except in the following special case: $d = 1$, $n = k$ is even, $|H| = 2$, and $K$ is a transitive subgroup of $C_2 \wr S_{n/2}$. As a simple application, we prove results about inclusions between wreath product subgroups., Comment: to appear in J. Algebra

Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on $\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of $u$ on $S^2(V)$., Comment: to appear in Linear Algebra and its Applications

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$., Comment: to appear in J. Pure Appl. Algebra

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As a part of this result, we describe the conjugacy classes of unipotent elements of $\operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$ in $\operatorname{Sp}(V_1 \otimes V_2)$., Comment: to appear in J. Algebra

Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known., Comment: to appear in Proc. Amer. Math. Soc

Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p \geq 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a, a' \in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a'$ are semisimple, and also proved the conjecture when $p = 0$. In this paper, we give a counterexample to Steinberg's conjecture. Specifically, we show that when $p = 2$ and $G$ is simple of type $C_5$, there exist two non-conjugate unipotent elements $u, u' \in G$ such that $f(u)$ and $f(u')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$., Comment: to appear in Transform. Groups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G, p) = (C_2, 2)$, the subgroup $X$ cannot be contained in a proper parabolic subgroup of $G$. In the case where $u$ has order $> p$, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight., Comment: 33 pages

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $\lambda$. When $V$ is self-dual, a basic question to ask is whether $V$ has a non-degenerate $G$-invariant alternating bilinear form or a non-degenerate $G$-invariant quadratic form. If $p \neq 2$, the answer is well known and easily described in terms of $\lambda$. In the case where $p = 2$, we know that if $V$ is self-dual, it always has a non-degenerate $G$-invariant alternating bilinear form. However, determining when $V$ has a non-degenerate $G$-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where $G$ is of classical type and $\lambda$ is a fundamental highest weight $\omega_i$, and in the case where $G$ is of type $A_l$ and $\lambda = \omega_r + \omega_s$ for $1 \leq r < s \leq l$. We also give a solution in some specific cases when $G$ is of exceptional type. As an application of our results, we refine Seitz's $1987$ description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If $X < Y < \operatorname{SL}(V)$ are simple algebraic groups and $V \downarrow X$ is irreducible, then one of the following holds: (1) $V \downarrow Y$ is not self-dual; (2) both or neither of the modules $V \downarrow Y$ and $V \downarrow X$ have a non-degenerate invariant quadratic form; (3) $p = 2$, $X = \operatorname{SO}(V)$, and $Y = \operatorname{Sp}(V)$., Comment: 46 pages; to appear in J. Algebra