Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic?

Question

I'm learning homomorphism and isomorphism in basic group theory. Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic? I couldn't think of one.

Answer 1

Lots of "natural" homomorphisms of symmetry groups arise from considering sets which they act on. For example, the unit square $[-1,1]\times [-1,1] \subseteq \mathbb R^2$ has a symmetry group $D_8$ of order $8$, consisting of rotations by $0,\pi/2,\pi$ and $3\pi/2$, and then four reflections -- two about the lines through the midpoints of opposite sides, and two about the lines through the vertices of the square which are not adjacent to each other.

Now one way to obtain a homomorphism from $D_8$ to another group is to consider sets on which $D_8$ acts: for example, $D_8$ acts on the four vertices of the square. This gives you an injective (but not surjective) group homomorphism $a\colon D_8 \to S_4$, where $S_4$ denotes the group of permutations of a 4-element set.

You can obtain a different homomorphism from $D_8$ to $S_4$ by considering the action of $D_8$ on the 4 reflecting lines, and if you draw the regular octagon which shares half of its vertices with the square, then it is easy to see that $D_8$ is a subgroup of $D_{16}$ the symmetries of a regular octagon.