When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E\_7(q))\cong Z\_2$ a square element in $2.E\_7(q)$?


A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is defined satisfying the following axioms:


(i) $[xy]$ is bilinear for $x, y\in L$.


(ii) $[xx]=0$ for $x\in L$.


(iii) $[[xy]z]+[[yz]x]+[[zx]y]=0$ for $x, y, z\in L$.


for each element $x$ of a Lie algebra $L$ we define a map ${\rm ad}~x$ of $L$ into itself by 
$${\rm ad}~x.y=[xy],~~~y\in L.$$


For each $x,y\in L$ we define the scalar product $(x,y)=tr(ad~x.ad~y)$, which is called the Killing form.


The dimension of the Cartan subalgebras $H$ of $L$ is called the rank of $L$, and will usually be denoted by $l$.


Although the roots are defined as elements of the dual space of $H$ they can, by considering the Killing form, be regarded as elements of $H$ itself.


Each element of the dual space of $H$ is expressible in the form $h\rightarrow (x, h)$ for a unique element $x\in H$. The element $x$ is associated with the map $h\rightarrow r(h)$ may be identified with the root $r$. Thus $r$ can be regarded either as an element of $H$ or an element of its dual space; the relation between these two being given by 
$$r(h)=(r, h),~~~h\in H.$$


We now define the Dynkin diagram of the Lie algebra $L$. This is a graph with $l$ nodes, one associated with each fundamental root $p\_i$, such that the $i$th node is joined to the $j$th node by a bond of strength $n\_{ij}$.


The Dynkin diagram of simple Lie algebra $E\_7$ is as follows:


Let $L$ be a Lie algebra over a field of characteristic $0$ and $\delta$ be a derivation of $L$ which is nilpotent, i.e. satisfies $\delta^n=0$ for some $n$. Then
$${\rm exp}~\delta=1+\delta+\frac{\delta^2}{2!}+...+\frac{\delta^{n-1}}{(n-1)!}$$
is an sutomorphism of $L$.


We write $x\_r(\zeta)={\rm exp}(\zeta ad~e\_r)$ for $\zeta\in \mathbb{C}$.


We shall write $h\_r$ for $\bar{h}\_r$, $e\_r$ for $\bar{e}\_r$, $x\_r(t)$ for $\bar{x}\_r(t)$, and $A\_r(t)$ for $\bar{A}\_r(t)$. This omission of the bars will not lead to confusion or inconsistency since the objects originaly called $h\_r$, $e\_r$, $x\_r(t)$, $A\_r(t)$ are special cases of $\bar{h}\_r$, $\bar{e}\_r$, $\bar{x}\_r(t)$, $\bar{A}\_r(t)$ when $K=\mathbb{C}$.


The Chevalley group of type $L$ over the field $K$, denoted by $L(K)$, is defined to be the group of automorphisms of the Lie algebra $L\_K$ generated by the $x\_r(t)$ for all $r\in \Phi$, $t\in K$.


We now consider the special case in which the base field $K$ is the finite field $GF(q)$ with $q$ elements, where $q$ is an arbitrary prime power. $G$ is then a group of non-singular linear transformations of a space over a finite field, so is a finite group. The Chevalley group of type $L$ over $GF(q)$ will be denoted by $L(q)$. 


$|E\_7(q)|=q^{63}(q^18-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^8-1)(q^6-1)(q^2-1)/{(2,q-1)}.$


The points over a finite field with $q$ elements of the (split) algebraic group $E\_7$, whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a fnite Chevalley group. This closely connected to the group written $E\_7(q)$, however there is ambiguity in this notation, which can stand for several thing:


1. The finite group consisting of the points over $F\_q$ of the simply connected form of $E\_7$ (for clarity, this can be written $E\_{7, sc}(q)$ and is known as the "universal" Chevalley group of type $E\_7$ over $F\_q$)
2. (rarely) the finite group consisting of the points over $F\_q$ of the adjoint form of $E\_7$ (for clarity, this can be written $E\_{7, ad}(q)$, and is known as the "adjoint" Chevalley group of type $E\_7$ over $F\_q$), or
3. the finite group which is the image of the natural map from the former to the latter: this is what will be denoted by $E\_7(q)$ in the following, as is most commmon in texts dealing with finite groups.


$E\_7(q)$ is simple for any $q$ and $E\_{7,sc}$ is its Schur cover, and we often write $E\_{7,sc}(q)$ as $2.E\_7(q)$ when $q$ is odd.


$\bar{K}$ denotes a semisimple algebraic group, with maximal torus $\bar{T}$ and root system $\sum$.


If $\sum=E\_7$, then genertors of $Z(\bar{K})$ are $h=h\_{\alpha\_4}(-1)h\_{\alpha\_5}(-1)h\_{\alpha\_7}(-1)$.


if $\sum=D\_{2m}$, then the generators of $Z(\bar{K})$ are $h\_1=h\_{\alpha\_1}(-1)h\_{\alpha\_3}(-1)...h\_{\alpha\_{2m-1}}(-1)$ and $h\_2=h\_{\alpha\_{2m-1}}(-1)h\_{\alpha\_{2m}}(-1)$


For a $\mathbb{C}G$-module $V$ with irreducible character $\chi$ we have the Frobenius-Schur indicator $$\nu(\chi) = \frac{1}{|G|} \sum\_{g \in G} \chi(g^2),$$


and $\nu(\chi)$ takes one of the values {+1, -1, 0}, as $\chi$ is afforded by a real representation or is real-valued but not afforded by a real representation or is not real-valued, respectively. 


Theorem 12.1.1 Let $L$ be a simple Lie algebra with $L\neq A\_1$ and let $K$ be a field. For each root $r$ of $L$ and each element $t$ of $K$ introduce a symbol $\bar{x}\_r(t)$. Let $\bar{G}$ be the abstract group generated by the elements $\bar{x}\_r(t)$ subject to relations
$$\bar{x}\_r(t\_1)\bar{x}\_r(t\_1)=\bar{x}\_r(t\_1+t\_2),$$
$$[\bar{x}\_s(u),\bar{x}\_r(t)]=\prod\_{i,j>0}\bar{x}\_{ir+js}(C\_{ijrs}(-t)^iu^j),$$
$$\bar{h}\_r(t\_1)\bar{h}\_r(t\_2)=\bar{h}\_r(t\_1t\_2),~~~t\_1t\_2\neq 0,$$
and 
$$\bar{n}\_r(t)=\bar{x}\_r(t)\bar{x}\_{-r}(-t^{-1})\bar{x}\_{r}{(t)}.$$
Let $\bar{Z}$ be the centre of $\bar{G}$. Then $\bar{G}/{\bar{Z}}$ is isomorphic to the Chevalley group $G=L(K)$.


Let $S$ be a Sylow 2-subgroup of $E\_7^u(q)$ (universal Chevalley group). Then $$Z(S)=\langle h\_e(-1), h\_{s\_3}(-1)h\_{s\_5}(-1)h\_{s\_7}(-1), h\_{s\_2}(-1)h\_{s\_3}(-1)\rangle.$$ Since 
$$h\_e(-1)=h\_{s\_2}(-1)h\_{s\_5}(-1)h\_{s\_7}(-1),$$ we conclude that 
$$Z(S)=\langle h\_{s\_3}(-1)h\_{s\_5}(-1)h\_{s\_7}(-1), h\_{s\_2}(-1)h\_{s\_3}(-1)\rangle\cong C\_2\times C\_2.$$
Recall that the center of $Z(E\_7^u(q))$ is $Z\_0=Z(E\_7^u(q))=\langle h\_{s\_3}(-1)h\_{s\_5}(-1)h\_{s\_7}(-1)\rangle$. It follows that $\bar{S}=S/{Z\_0}$ is a Sylow $2$-subgroup of $E\_7(q)$.


The center of $D\_6^u(q)$ is $\langle h\_{s\_3}(-1)h\_{s\_5}(-1)h\_{s\_7}(-1), h\_{s\_2}(-1)h\_{s\_3}(-1)\rangle$.


$D\_m(q)\cong P\Omega\_{2m}^+(q)$ for $m\geq 3$.


If $n=2m$ and $q^m\equiv -\epsilon~{\rm mod}~4$, then $\Omega\_n^\epsilon(q)$ is already simple, and the spin group has the structure $2\cdot \Omega\_n^\epsilon (q)$. If $n=2m$ and $q^m\equiv \epsilon~{\rm mod}~4$, then $\Omega\_n^\epsilon(q)$ has a centre of order $2$, and the spin group has the structure $4.P\Omega\_n^\epsilon(q)$ if $m$ is odd, and the structure $2^2.P\Omega\_n^{\epsilon}(q)~$(necessarily with $\epsilon +$) if $m$ is even.


When $m$ is an even integer and $q$ a power of a odd prime, then $q^m\equiv 1~{\rm mod~4}$.


Analysis:


$$\pi: 2.E\_7(q)\rightarrow E\_7(q)$$


The following websites may be useful to my quesion:


<https://math.stackexchange.com/questions/785603/what-do-sylow-2-subgroups-of-finite-simple-groups-look-like>


[Kernel of a double cover of group as stem extension](https://mathoverflow.net/questions/273929/kernel-of-a-double-cover-of-group-as-stem-extension)


[Square roots of elements in a finite group and representation theory](https://mathoverflow.net/questions/42646/square-roots-of-elements-in-a-finite-group-and-representation-theory?rq=1)



Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
The answer is always, yes. Note that there are three classes of involutions in the simply connected version of the algebraic group $E\_7$: the central involution $a$, an involution $t$ with centralizer of type $A\_1D\_6$, and the product $at$. If $a$ were not a square, then in the simple group $E\_7(q)$, we would only see involutions with centralizer type $A\_1D\_6$. However, in the adjoint group we find centralizers of type $E\_6T\_1$ and $A\_7$. This website: <http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/23elts.html> lists the classes of involutions in the simply connected and adjoint groups, or consult the 3rd volume of Gorenstein-Lyons-Solomon.


Notice that this yields explicit elements of the simple group that power to the centre in the central extension.


