Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} x H x^{-1}$ is connected

Question

Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} x H x^{-1}$ is connected.

My attempt.

I constructed the function $g_x : H \to S$ such that $g_x(h) = x h x^{-1}$. This function is continuous thus $g_x(H)$ is connected.

I believe, $g_x(H) \cap g_y(H) = \emptyset$ for $x \neq y$ and $x,y \notin H$, from this way we can conclude that $S$ is connected but I'm not sure.

Any help will be appreciated.

You want (and have) $1\in g_x(H)\cap g_y(H)\color{red}\ne\emptyset$. — Hagen von Eitzen, May 01 '19 at 09:14

score 1 · Accepted Answer · answered May 01 '19 at 09:14

1

For each $x\in G$, $e\in xHx^{-1}$. So, $\{xHx^{-1}\,|\,x\in G\}$ is a set of connected sets with non-empty intersection. Therefore, its union is connected.

answered May 01 '19 at 09:14

José Carlos Santos

440,053

Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} x H x^{-1}$ is connected

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