My Math Forum prove that Ax(e) is a subgroup of AxB

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 February 7th, 2012, 10:56 PM #1 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 prove that Ax(e) is a subgroup of AxB Can someone explain to me how to do the following? Prove that $A \times \{e\}$ is a subgroup of $A\times B$ I know that we should apply the subgroup criterion but what elements does $A \times \{e\}$ have???
 February 8th, 2012, 12:54 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: prove that Ax(e) is a subgroup of AxB The elements of A x {e} are {a, e} where a is an element of A and e is the identity element of B. So if A is Z_3 and B is Z_7, A x {e} is {(0, 1), (1, 1), (2, 1)} since 1 is the identity in Z_7 and the elements of Z_3 are 0, 1, and 2.

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