A3 in A5

Let $$G$$ be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set $$\{ 1,2,3,4,5 \}$$. $$G$$ has order $$5!/2 = 60$$.

Consider the subgroup:

$$H = H_{\{1,2,3 \}} = \{, (1,2,3), (1,3,2) \}$$

$$H$$ has a total of 10 conjugate subgroups (including $$H$$ itself) and the subgroups are parametrized by subsets of size 3 in $$\{1,2,3,4,5 \}$$ describing the support of the 3-cycles. The complementary subset of size two is fixed point-wise by that conjugate subgroup:

Each of these subgroups is isomorphic to cyclic group:Z3.

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

Sylow and corollaries
The subgroup is a 3-Sylow subgroup, so many properties follow as a corollary of that.