Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Suppose $x=ab,a\in H\times 1,b\in 1\times K$
Then $x=(h,1)(1,k)$ where $h\in H,k\in K$
Hence $x=(h,k)\in H\times K$

Let $(h,k)\in H\times K$
Then $(h,k)=(h,1)(1,k)\in (H\times 1)(1\times K)$

What I want to ask is $(h,1)(1,k)=(h,k)$ always true and is there any difference between $(H\times 1)(1\times K)$ and $(H\times 1)\times(1\times K)$?

1 Answer

Multiplication in $H\times K$ is given by $$(h_1,k_1)(h_2,k_2)=(h_1h_2,k_1k_2)$$ so, yes, $(h,1)(1,k)=(h1,1k)=(h,k)$.

Now, elements of $H\times K=(H\times 1)(1\times K)$ are pairs $(h,k)$ with $h\in H$ and $k\in K$. Elements of $(H\times 1)\times(1\times K)$ are pairs of pairs $((h,1),(1,k))$ with $h\in H$ and $k\in K$. Of course, there is an obvious isomorphism $$H\times K\to (H\times 1)\times (1\times K)$$ so this difference is superficial.