Use a lemma to prove that A4 has no subgroup of order 6
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Use a lemma to prove that A4 has no subgroup of order 6

[From: ] [author: ] [Date: 13-10-31] [Hit: ]
So the elements of A4 arent all showing up.in your question.A4 is the set of even permutations, which can be represented as an even number of 2-cycles.(1), (12)(34),......
Lemma: If H≤G has index 2, i.e. [G:H]=2, then for any a∈G we have a^2∈H.

The 12 elements of A4 are (1),(12)(34),(13)(24),(14)(23),(123),(13… (234), and (243).

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If you have a long string with no spaces, Yahoo Answers truncates the string. So the elements of A4 aren't all showing up.in your question.

A4 is the set of even permutations, which can be represented as an even number of 2-cycles. They are
(1), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143), (234), (243)

Suppose H is a subgroup of A4 of order 12.

Let's look at a^2 for a in A4
(1) gives (1)
(12)(34) gives (1)
(13)(24) gives (1)
(14)(23) gives (1)
(123) gives (132)
(132) gives (123)
(124) gives (142)
(142) gives (124)
(134) gives (143)
(143) gives (134)
(234) gives (243)
(243) gives (234)

If a is in G, a^2 is in G. If a is not in G, a^2 is still in G, by the lemma. Therefore all the squares in the above list have to be in G. The trouble is, there are 9 element in the list. Since 9 > 6, there is no subgroup of order 6.
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