Check if is a retract of . Proof: Define by for all . Let . Then where are integers. So, is homomorphism. Clearly, for all . Since , . Hence, need not be a retract of .
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Originally Posted by deniselim17 Check if is a retract of . Proof: Define by for all . Let . Then where are integers. So, is homomorphism. Clearly, for all . Since , . Hence, need not be a retract of . What is t, anyway? A normal subgroup of a group is a retract of the group iff it has a normal complement. Tonio
is a cyclic group generated by . is a generator.
Originally Posted by deniselim17 is a cyclic group generated by . is a generator. Good. So <t> is a cyclic group and thus it cannot be expressed as a non-trivial direct product ==> no subgroup of <t> has normal complement ==> there exists no retract of <t>. Tonio
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