More specifically:
Let G1 be a subgroup of the group G.
Let H1 be a subgroup of the group H.
Prove G1 x H1 is a subgroup of G x H.
I've already shown the identity element e is in G1 x H1,
and I've already shown closure under the * operator.
However, I don't know how to prove that for every element in G1 x H1,
the inverse is an element as well, which I know is the last step to proving what was needed.
Best Answer
Let $(g,h) \in G_{1}\times H_{1}$.
Then $g \in G_{1}$ and $h \in H_{1}$.
Then $g^{-1} \in G_{1}$ and $h^{-1} \in H_{1}$ (since they are subgroups, and therefore all elements have inverses).
Therefore $(g^{-1},h^{-1}) \in G_{1}\times H_{1}$.