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Crossed modules are given by the data $(H \xrightarrow{t} G,\alpha: G \times H \to H)$ where $\alpha$ is an action and $t$ is a group homomorphism between a pair of groups, satisfying two conditions: equivariance and Pieffer identity. See the full definition here. We will denote $\alpha(g,h) = g\cdot h$.

In all the examples listed in the wikipedia page, the following equation is satisfied: $$g \cdot e_H = e_H,$$ where $e_H$ is the identity of $H$. It is easy to show this if $t$ is an injection or a surjection. I don't know if this statement is true in general.

So the question is:

Prove or disprove: $g \cdot e_H = e_H$ for any $g \in G$ in a crossed module.

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Judging by the definition of wikipedia, if $G$ acts on $H$ by automorphisms this means that for any $g\in G$ the function $g\cdot -:H\rightarrow H$ is a group-automorphism. In particular it is a group-homomorphisms, sending the identity element $e_H\in H$ to itself. In other words $g\cdot e_H = e_H$ for any $g\in G$.

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