Presentation of the trivial group

Dumb question: Is $$\langle e \mid e\rangle$$ a presentation of the trivial group?

• What do you mean by e on the right? Is it like e=e or e=e^2? Commented Dec 29, 2021 at 12:52
• Yes. ${}{}{}{}$ Commented Dec 29, 2021 at 13:00
• $e=e$, I guess. It's a trivial relation. Commented Dec 29, 2021 at 13:04
• @DietrichBurde $\langle e \mid \rangle$ is the infinite cyclic group, not the trivial group. Commented Dec 29, 2021 at 14:17
• The trivial group can be generated by taking no generators, subject to no relations: $\langle\,\vert\,\rangle$. Commented Dec 29, 2021 at 15:19

Yes. What exactly $$\langle X|R\rangle$$ means? We start with a free group $$F(X)$$ on $$X$$ and then we take the quotient $$F(X)/N(R)$$ where $$N(R)$$ is the normalizer of some subset $$R\subseteq F(X)$$.
So what is $$\langle e| e\rangle$$? While the "$$e$$" notation may be misleading (a typical symbol for identity) this is the same as $$\langle g|g\rangle$$, just by simple relabelling. And formally this is the same as $$\langle \{g\}|\{g\}\rangle$$ which by definition is the same as $$F(g)/N(g)$$. But since $$g$$ generates $$F(g)$$ and $$\langle g\rangle\subseteq N(g)$$ then $$N(g)=F(g)$$. And so the quotient $$F(g)/N(g)$$ is trivial.
Side note: When we write presentations, e.g. $$\langle x,y\ |\ xy\rangle$$ we often write it in a more readable way, e.g. $$\langle x,y\ |\ xy=e\rangle$$. And so we introduce the neutral element "$$e$$" symbol to see it better, but formally we don't actually need this neutral element directly. And so care has to be taken when you use the same symbol for a generator and the neutral element. Ideally you would never do that. Otherwise you are likely to run into problems, for example we can write $$\langle g|g\rangle$$ as $$\langle g|g=e\rangle$$ but we cannot write $$\langle e|e\rangle$$ as $$\langle e|e=e\rangle$$ because of the ambiguity. But we can write $$\langle e|e\rangle$$ as $$\langle e|e=1\rangle$$ if we denote the neutral element by "$$1$$". However you won't see such presentations often, it doesn't feel natural.