# Deck Transformations Question

Using the Hatcher's notation on proposition 1.39. Let $$G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$$ be the Deck transformation group for the covering space $$p: \tilde{X} \to X$$.

Question: Could someone please explain to me why $$G(\tilde{X})\cong \pi_1(X,x_0)/p_*(\pi_1(\tilde{X},\tilde{x_0}))$$ holds just when the covering $$\tilde{X}$$ is a normal covering? Please provide a counterexample when this does not hold when the covering is not normal.

Many thanks.

• If the covering is not normal, how can you talk about a group isomorphism? – Connor Malin Mar 9 at 12:21
• Such a simple but really precise and correct answer. Thanks! – rarc Mar 10 at 8:31