(a) Explanation Let D 4 = { e , α , α 2 , α 3 , β , γ , Δ , θ } be the octic group of order 8 . Let f be a permutation of set A . For each a ∈ A , define the stabilizer of a as stab ( a ) = { f ∈ S ( A ) | f ( a ) = a } . The elements of the group D 4 are as follows: e = { e ( 1 ) = 1 e ( 2 ) = 2 e ( 3 ) = 3 e ( 4 ) = 4 , α = { α ( 1 ) = 2 α ( 2 ) = 3 α ( 3 ) = 4 α ( 4 ) = 1 , α 2 = { α 2 ( 1 ) = 3 α 2 ( 2 ) = 4 α 2 ( 3 ) = 1 α 2 ( 4 ) = 2 α 3 = { α 3 ( 1 ) = 4 α 3 ( 2 ) = 1 α 3 ( 3 ) = 2 α 3 ( 4 ) = 3 , β = { (b) To determine The stabilizer of ( 2 ) for the symmetric group D 4 (c) To determine The stabilizer of ( 3 ) for the symmetric group D 4 (d) To determine The stabilizer of ( 4 ) for the symmetric group D 4

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Chapter 4.1, Problem 28E

(a)

To determine

The stabilizer of (1) for the symmetric group D4

(b)

To determine

The stabilizer of (2) for the symmetric group D4

(c)

To determine

The stabilizer of (3) for the symmetric group D4

(d)

To determine

The stabilizer of (4) for the symmetric group D4

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Chapter 4.1, Problem 27E

Chapter 4.1, Problem 29E

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The stabilizer of ( 1 ) for the symmetric group D 4

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The stabilizer of (1) for the symmetric group D4

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