Thread: Conjugacy Class of Elements of Order 4 in S8 and S12

1. Conjugacy Class of Elements of Order 4 in S8 and S12

Dummit and Foote Section 4.3 Exercise 12 states:

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Find a representative of each conjugacy class of elements of order 4 in $\displaystyle S_8$ and in $\displaystyle S_{12}$.
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I am lost as to how to approach this problem.

Peter

2. Re: Conjugacy Class of Elements of Order 4 in S8 and S12

well, first we have to have a way of describing the elements of order 4. let's tackle S8 first.

clearly any 4-cycle is of order 4. since 4 = 2*2, the disjoint cycle decomposition of an element of S8 of order 4, can only contain 2-cycles and 4-cycles.

that gives the following possibilities:

(a b c d) a 4-cycle
(a b c d)(e f) a disjoint 4-cycle and 2-cycle
(a b c d)(e f)(g h) a 4-cycle and 2 disjoint 2-cycles
(a b c d)(e f g h) 2 disjoint 4-cycles.

in Sn, (disjoint) cycle types determine the conjugacy classes. that is two permutations with the same disjoint cycle types are conjugate, and vice-versa. can you tackle S12, now?

3. Re: Conjugacy Class of Elements of Order 4 in S8 and S12

Thanks for that Deveno - most helpful

Reckon I can tackle S12 OK now

Peter

conjugacy classes of s12

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