How many homomorphisms are there from \[ \mathbb Z_2\times\mathbb Z_2 \] to the symmetric group on three objects?
Solution
Write \[ V=C_2\times C_2 = \gens{a,b}, \] where \[ a^2=b^2=e, \qquad ab=ba. \]
1. A homomorphism \(\varphi\colon V\to S_3\) is determined uniquelyby an ordered pair \((x,y)\) of commuting elements of \(S_3\) satisfying\[ x^2=y^2=e. \]
The images \[ x=\varphi(a), \qquad y=\varphi(b) \] must satisfy the defining relations of \(V\), so \(x^2=y^2=e\) and \(xy=yx\). Conversely, any ordered pair satisfying these relations determines a unique homomorphism from the displayed presentation of \(V\).
2. The elements of \(S_3\) whose square is the identity are theidentity and the three transpositions.
The elements of \(S_3\) have cycle types \[ 1,\qquad (12),\qquad (123). \] The identity has order \(1\), the three transpositions have order \(2\), and the two \(3\)-cycles have order \(3\). Thus precisely the identity and the three transpositions square to the identity.
3. There are exactly \(10\) ordered commuting pairs among the fourelements from step
2.
If \(x=e\), then \(y\) may be any of the four elements from step 2, giving \(4\) choices.
If \(x\) is a transposition, then among those four elements it commutes only with \(e\) and itself. Indeed, two distinct transpositions in \(S_3\) share one letter; after relabeling they are \((12)\) and \((13)\), and \[ (12)(13)=(132)\neq(123)=(13)(12). \] Thus each of the three possible transpositions for \(x\) gives \(2\) choices for \(y\). The total number is therefore \[ 4+3\cdot2=10. \]
4. The requested number of homomorphisms is\[ \boxed{10}. \]
By step 1, homomorphisms are in bijection with the ordered pairs counted in step 3.
5. Q.E.D.
Step 4 is the required count.