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Moderate

Presentation of a Group: Generators and Relations (Relators)

ABSALG-LXTKN3

Let $F=F(B)$ be a free group with basis $B$ and let $R$ be a set of reduced words on $B$.

A group $G$ is said to be the group defined by generators $b\in B$ and relations (or relators) $r\in R$, when:

$G=F/N$ where $N=N(R)$ is the normal subgroup of $F$ "generated by $R$".

The subgroup $N$ is the intersection of all normal subgroups of $F$ that contain $R$.

We write $G=\langle B\mid r=1, r\in R\rangle$.

Notation: $A_n$ denotes the alternating group (the subgroup of even permutations in $S_n$), $n\ge 2$.

Which of the following are TRUE?

Select ALL that apply.

A

The group $\langle a,b\mid a^2=b^2=(ab)^2=1\rangle$ has order $8$

B

The additive group ${\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$ is of the form $F/N(R)$ for
$F=F(a,b)$ and $R=\{a^2,b^2,(ab)^2\}$

C

$F(a)/N(R)$, where $R=\{a^2\}$ can be realized as a subgroup of $\langle a,b\mid a^2=b^2=(ab)^2=1\rangle$.

D

The group $F(a,b)/N(R)$, where $R=\{a^2,b^2,(ab)^2\}$ is contained in the subgroup $A_3$ of $A_4$.

E

$\langle a,b\mid a^2=b^2=(ab)^2=1\rangle$ is a subgroup of $F(a,b)/N(R)$ where $R=\{a^2, b^2, (ab)^3\}$.