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1. Let G be a group of order 4. Prove that every element of G has order 1,2, or 4.
2. Classify groups of order 4 by considering the following two cases:
a. G contains an element of order 4
b. Every element of G has order <4
could anyone please help me to solve these problems?
and what does it mean by 'Classify groups of order 4'?
the group itself will have order 4,|e|=1
for any element a, |a|=1 or 2 or 4
if |a|=1 then a=e
if there is an a such that |a|=4, than it would be a cyclic, thus isomorphic to Z(4), so it will also have subgroup of order 2.
if such element does not exist, then 3 elements will have order 2, that is, isomorphic to Z(2)*Z(2).
"classify"actually means classify the group up to isomorphism