a) Using Cayley’s theorem, find the permutation group to which a cyclic group of
order 12 is isomorphic. (4)
b) Let τ be a fixed odd permutation in . S10 Show that every odd permutation in S is 10
a product of τ and some permutation in . A10 (2)
c) List two distinct cosets of < r > in , D10 where r is a reflection in . D10 (2)
d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer.
a) Given a cyclic group G = <a> of order 12 generated by the element 'a’ of G, we may take the 'permutation group’ as the cyclic subgroup H of S generated by the cycle (1,2,3,4,5,6,7,8,9,10,11,12) of length 12. This (or a conjugate of it) is what we obtain as per the proof of the Cayley's Theorem.
We may also find isomorphic copies of G as subgroups of groups S for all n 7. For example the product (1,2,3,4)•(5,6,7) of disjoint cycles in S has order LCM(4,3)=12 and hence generates a cyclic subgroup of S , isomorphic to G.
d)
has order and is not abelian if
Answer:
c)
Let
two cosets:
b)
the subset of even permutations in .
is product of odd number of transpositions in ; permutation in is product of even number of transpositions in . So, result of product of and permutation in is:
(odd number of transpositions in )+(even number of transpositions in )=
=(odd number of transpositions in ), that is odd permutation in .
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