Recall that |S_4| = 24. All of the elements can be written in cycle notation as
ID: 1720155 • Letter: R
Question
Recall that |S_4| = 24. All of the elements can be written in cycle notation as the identity, a cycle of length 2, a cycle of length 3, a cycle of length 4 or a product of two disjoint cycles of length 2. The different types of elements are said to have cycle type and 2^2 respectively. [ Cycle type in symmetric groups is well-defined. We'll prove this later.] List all the possible orders of a subgroup of S_4 allowed by Lagrange's theorem. Show that psi is a homomorphism. Show that there is a subgroup of each order allowed by Lagrange's theorem.Explanation / Answer
|S4| has 4! elements that is 24
cycle length elements
1 (e)
2 (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)
3 (1,2,3),(1,2,4),(1,3,4),(2,3,4)(1,3,2),(1,4,2),,(1,4,3),(2,4,3)
4 (1,2,3,4),(1,3,4,2),(1,4,2,3),(4,3,2,1),(2,4,3,1),(3,2,4,1)
2,2 (1,2)(3,4), (1,3)(2,4),(1,4)(2,3)
a total of 24
now Lagrange’s Theorem states that Let G be a finite group, and let H be a subgroup of G. Then the order of H divides the order of G.
a)as |s4| = 24 the possible orders are 1,2,3,4,6, 12, 24.
c) as you see the subgroup orders are 1,2,3,4...these all are alowed by lagrange's theorem.
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