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Prove If a subgroup H of Sn contains an odd permutation, then H is

Prove If a subgroup H of Sn contains an odd permutation, then H is

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Prove: If a subgroup H of Sn contains an odd permutation,

then |H| is even and exactly half the elements of H are odd

permutations.

Solution

Take any subgroup H of S_n which is _not_

contained in A_n then since A_n is a normal subgroup of

S_n it follows that (H A_n)/A_n is isomorphic to H/(H

intersection A_n). Since H is not contained in A_n then H

contains an odd permutation so the product HA_n is in fact

equal to S_n. Therefore [H Int A_n: H] = 2 so H has even

order. Consider HnA(n) and HA(n). Use the fact that A(n)

is an index 2 normal subgroup of S(n). Even simpler, if H

contains an odd permutation \'o\', then H*o=H. But

multiplying by o turns even permutations into odd, and vice

versa. Thus exactly half elements are even and half are

odd permutations

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Prove: If a subgroup H of Sn contains an odd permutation, then |H| is even and exactly half the elements of H are odd permutations. Solution Take any subgroup H of S_n which is _not_ contained in A_n then since A_n is a normal sub group of S_n it follows that (H A_n)/A_n is isomorphic to H/(H inte ...
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Apply MANOVA and Prepare a Course Reflection

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Apply MANOVA and Prepare a Course Reflection

You will submit one Word document. You will create this Word document by cutting and pasting SPSS output into Word. Part A ...

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