# Identity component of a Lie group

Could any one help me to solve this problem?

Let the identity component $G_0$ of a Lie Group $G$ be the connected component of the identity element $e\in G$. Let $\mu$ and $i$ be the multiplication map and inverse map of $G$

1. For any $x_0\in G_0$ Show that $\mu(\{x_0\}\times G_0)\subseteq G_0$.

2. Show that $i(G_0)\subseteq G_0$

3. $G_0$ is an open subset of $G$.

4. prove that $G_0$ is a Lie Group.

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What have you tried? Also, for a) should $x_0$ be $x$? –  Alex Becker Aug 18 '12 at 20:56
I don't understand 2. Are you sure you want $\subsetneq$, not $\subseteq$ (in fact you have equality)? The first two properties immediately follow from connectedness and continuity of the maps involved. The fourth follows directly from 1.,2.,3. It remains to ponder 3. –  t.b. Aug 18 '12 at 20:59
yes, In the question paper it was proper inclusion –  Une Femme Douce Aug 18 '12 at 21:03
Then it is wrong. Was it $\subset$? Note that many people allow equality when writing $\subset$. Consider some examples, like $O(2)$ and $O(3)$, or whatever Lie group you understand well. –  t.b. Aug 18 '12 at 21:04
Or simply consider any connected Lie group to see that 2 is wrong. –  Alex Becker Aug 18 '12 at 21:05