Effect of Generators and Relations on a Finite Group's Order

Question

Say we define a group $G$ by generators and relations and have shown that $G$ is finite. If we then create a group $G'$ by imposing one additional relation on $G$ which is not implied by any of the original ones, then is it always true that the order of $G'$ is strictly less than $G$?

Answer 1

The answer is yes.

Formally, let $G=\langle X\mid R\rangle$ be a presentation for $G$; let $r\in F(X)$ be an element of the free group in $X$ such that $r$ is not an element of $\langle R\rangle^{F(X)}$; that is, a word that is not implied by the relations $R$. Let $S=R\cup\{r\}$. And let $G’=\langle X\mid S\rangle$.

Since $R\subseteq S$, the identity map $X\to X$ induces a map from $G$ onto $G’$, by von Dyck’s Theorem. This corresponds to the fact that if $N=\langle R\rangle^{F(X)}$ and $M=\langle S\rangle^{F(X)}$, then $G\cong F(X)/N$ and $G’\cong F(X)/M$, and $N\leq M$ so we have a natural map $G\to G’$. However, $r\in M\setminus N$, so the image of $r$ in $G$ is nontrivial and maps to the identity in $G’$. Thus, the kernel of the morphism $\varphi\colon G\to G’$ is nontrivial.

Since you are assuming that $G$ is finite, the nontriviality of the kernel implies that $|G’|$ is strictly less than $|G|$.