Definition.A finite group $G$ is calledsquared(resp.almost squared) if there exists a subset $A\subseteq G$ such that $G=\{ab:a,b\in A\}$ and $|G|=|A|^2$ (resp. $|G|=|A|^2-1$). Such a set $A$ will be called an (almost)square rootof $G$.

According to the answers to this MO-post, no nontrivial finite group is squared.

In contrast, nontrivial almost squared finite groups do exist. The simplest one is the 3-element cyclic group $C_3$. Any 2-element subset of $C_3$ is an almost square root of $C_3$.

A less trivial example is the dihedral group $D_8=\langle a,b\;|\;a^4=b^2=1,\;bab=a^3\rangle$ with almost square root $A=\{a,b,ba\}$.

Three other examples of almost squared groups (found by GAP) are:

$\bullet$ the symmetric group $S_4$;

$\bullet$ the general linear group $GL(2,3)$ of non-degenerate $2\times 2$ matrices over the 3-element field,

$\bullet$ the symmetric group $S_5$.

Problem 1.Find more examples of almost squared finite group. Are there infinitely many almost squared finite groups?

**Remark.** Using GAP, Voldymyr Gavrylkiv established that among groups of order $<168$ the only almost squared groups are the groups $C_3$, $D_8$, $S_4$, $GL(2,3)$, and $S_5$. Those groups have orders 3, 8, 24, 48, and 120, respectively. It is interesting that no almost squared group of order 80 exists.

Problem 2.What can be said about the structure of almost squared groups?

**Remark.** Alex Ravsky observed that for an almost square root $A$ of an almost squared group $G$, the center $Z(G)$ of $G$ is almost contained in the set $A^2=\{a^2:a\in A\}$ in the sense that $Z(G)\setminus A^2$ contains at most one element. So, $|Z(G)|\le|A|+1=1+\sqrt{|G|+1}$, which implies that the unique almost squared commutative group is $C_3$.

The only known (at the moment) almost squared noncommutative groups $D_8$, $S_4$ and $GL(2,3)$ have even cardinality.

Problem 3.Is the cardinality of any almost squared noncommutative group even?

**Remark.** It can be shown that all noncommutative groups of odd order $<675=3^3\times 5^2$ are not almost squared.

Problem 3'.Is there an almost squared group among groups of order $675$?

Problem 4.Let $A$ be an almost square root of an almost squared non-commutative finite group $G$. Are there distinct elements $a,b\in A$ such that $a^2=b^2=g$ for some $g\in G$ such that:

$\bullet$ $g\in Z(G)$?

$\bullet$ $g^2\in Z(G)$?

$\bullet$ $g^2=1$?

$\bullet$ $g=1$?

$\bullet$ $g$ has order $\le 3$?

**Remark.** For any almost square root in any of three known almost squared noncommutative groups $D_8,S_4,GL(2,3)$, there exist distinct elements $a,b\in A$ such that $a^2=b^2=1$. In these group the center contains at most two elements.

The following problem was suggested by @LSpice in his comment.

Problem 5.Let $G$ be an almost squared noncommutative group. Is $z^2=1$ for any central element of $G$?

Problem 4if some central translate of $A$ satisfies your condition—unless you are making the additional hypothesis that the centre of an almost squared group has exponent $2$? (One could perhaps also ask about how many almost square roots there are; surely it would be too much to expect that they are all central translates of one another.) $\endgroup$ – LSpice Feb 21 at 5:2610more comments