# Find $G/Z(G)$ given the following information about the group?

$$G$$ is a finite group generated by two elements $$a$$ and $$b$$, we are given the following data:

Order of a= $$2$$

Order of $$b=2$$

Order of $$ab=8$$.

If $$Z(G)$$ denotes the center then what is $$G/Z(G)$$ isomorphic to?

Attempt:

To be honest I don't know how to start this. I thought of taking $$D_8$$ as a concrete example of such a group but was not able to proceed much.

One thing I can see that this group is non abelian as if it was abelian then it would imply that $$(ab)^2=e$$ which contradicts the fact that order of $$ab$$ is 8.

$$G$$ is indeed the dihedral group of order $$16$$.

A useful fact: The center of dihedral group $$D_{2n}$$ (notation: $$D_{2n}$$ is the dihedral group of order $$2n$$) is trivial if $$n$$ is odd and is $$\pm 1$$ if $$n$$ even. This is easily seen from the relation $$r\bar{r}=\bar{r}r^{-1}$$.

For $$n>2$$, the quotient $$D_{2n}/Z$$ is also generated by two elements of order $$2$$, so is dihedral of the appropriate order.

• The quotient doesn't appear to be dihedral. – Chris Custer Jan 3 at 3:37
• @ChrisCuster Huh? It is dihedral from the relations (which automatically hold for quotients) $\bar{r}^2=(r\bar{r})^2=1$ and order. – user10354138 Jan 4 at 22:47
• Oops. My mistake. – Chris Custer Jan 5 at 0:48

In general, $$G/Z(G)\cong\operatorname{Inn}G$$, the group of inner automorphisms of $$G$$.

Since in this case $$G=D_{16}$$ (see this), and $$Z(D_{16})=\{1,r^4\}$$, we have $$\operatorname{Inn}G\cong D_{16}/\mathbb Z_2\cong D_8$$.