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## Compute the kernel of the left regular action of a group on itself

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.13

Find the kernel of the left regular action.

Solution: Recall that the left regular action of a group $G$ on itself is given merely by left multiplication. I claim that the kernel of this action is the trivial subgroup. Suppose $g$ is in the kernel; then for all $a \in G$, $ga = g \cdot a = a$. Right multiplying by $a^{-1}$ we see that $g = 1$, as desired.