# How to prove the connectedness or irreducibility of a variety?

Let $G$ be an algebraic group, and $\mu$ is the multiplication in $G$. Define a morphism $G \times G \times G \times G \rightarrow G \times G$ by $\mu \times \mu$, and let $X$ be the inverse image of the diagonal $\{(x,x)|x \in G\}$. If $G$ is connected, prove that $X$ is a closed irreducible subset of $G \times G \times G \times G$.

The closedness part of this proof is clear. But I have no idea as to the connectedness or the irreducibility of $X$.

A morphism between two varieties is continuous, and the multiplication, as well as inverse map of an algebraic group are all morphisms. These can be applied to determine whether a set is closed. But, what are the efficient ways to prove connectedness?

Many thanks~

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Notice that $X$ is isomorphic to $G\times G\times G$ ($X$ is composed of the quadruples $(a,b,c,d)$ such that $ab=cd$, and the isomorphism $X\to G^3$ is given by $(a,b,c,d)\to (a,b,c)$ ($d=c^{-1}ab$)). – user8268 May 20 '11 at 13:01
Thank you very much for this useful hint~ $X$ is closed and connected as $G^3$ is~ – ShinyaSakai May 21 '11 at 8:57