- #1
TheForumLord
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Homework Statement
The problem is:
Let G be a group of order 12 ( o(G)=12).
Let's assume that G has a normal sub-group of order 3 and let a be her generator ( <a>=G ).
In the previous parts of the questions I've proved that:
1. a has 2 different conjucates in G and o(N(a))=6 or o(N(a))=12 (N(a) is the normalizer of a)
2. there is a b in G of order 2 who is commutative with a: ab=ba
3. if o(N(a))=6 then N(a) is a group of order 6 that is generated by ab ( <ab>=N(a) )
I have no clue in the next 2 parts of the question:
4. Prove that b is in the center of the group G (C(G)) . There's a clue: notice that b is in
N(a).
5. prove that if 2 doesn't divide o(C(G)) then G is isomorphic to A4 (A4 is the group of all even permutations)...
Homework Equations
None...
The Attempt at a Solution
In question 5 , I can't really understand how the two groups can be isomorphic if A4 has order 13 and G has order 12...
There's a clue to question 5 too: you can make use of Cayle's theorem...
I will be delighted if someone will help me...I have no clue in 4 and 5... detailed answers will be received with happines...
TNX!