Direct product of groups is essentially associative

Let A, B, and C be groups. Show that A \times (B \times C) \cong (A \times B) \times C.


We know from set theory that the mapping \varphi : A \times (B \times C) \rightarrow (A \times B) \times C given by \varphi((a,(b,c))) = ((a,b),c) is a bijection, with two-sided inverse \psi((a,b),c) = (a,(b,c)). Moreover \varphi is a homomorphism, as we show.

Let a_1, a_2 \in A, b_1,b_2 \in B, and c_1,c_2 \in C. Then \varphi((a_1,(b_1,c_1)) \cdot (a_2,(b_2,c_2))) = \varphi((a_1a_2, (b_1,c_1)\cdot(b_2,c_2))) = \varphi((a_1a_2, (b_1b_2, c_1c_2))) = ((a_1a_2, b_1b_2),c_1c_2) = ((a_1,b_1) \cdot (a_2,b_2), c_1c_2) = ((a_1,b_1),c_1) \cdot ((a_2,b_2),c_2) = \varphi((a_1,(b_1,c_1))) \cdot \varphi((a_2,(b_2,c_2))).

Thus \varphi is an isomorphism.

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