Conjugation is a group action

Let G be a group. Show that the mapping defined by g \cdot a = gag^{-1} does satisfy the axioms of a left group action of G on itself. (Called conjugation.)


We have 1 \cdot a = 1a1^{-1} = a. If g_1, g_2 \in G, then g_1 \cdot (g_2 \cdot a) = g_1 \cdot g_2ag_2^{-1} = g_1g_2ag_2^{-1}g_1^{-1} = (g_1g_2)a(g_1g_2)^{-1} = (g_1g_2) \cdot a. Thus this mapping is a group action.

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