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Chapter 2
#1 The Definition of Group
1
(1)
1 = (1 2 3), x = (2 3 1), x^2 = (2 3 1)(2 3 1) = (3 1 2),
y = (2 1 3), xy = (2 3 1)(2 1 3) = (1 3 2), x^2y = (3 1 2)(2 1 3) = (3 2 1)
(2)
1 | x | x^2 | y | xy | x^2y | |
---|---|---|---|---|---|---|
1 | 1 | x | x^2 | y | xy | x^2y |
x | x | x^2 | 1 | xy | x^2y | y |
x^2 | x^2 | 1 | x | x^2y | y | xy |
y | y | x^2y | xy | 1 | x^2 | x |
xy | xy | y | x^2y | x | 1 | x^2 |
x^2y | x^2y | xy | y | x^2 | x | 1 |
2
(a)
Associativity: By the associativity of matrix multiplication.
Identity: The identity matrix.
Inverse: The inverse matrix. Every matrix in has an inverse matrix since it's determinant is not zero.
Closure: The product of two invertible matrix is also invertible.
(b)
Associativity: By the associativity of maps.
Identity: The identity permutation.
Inverse: The inverse permutation.
Closure: By definition of
3
Associativity: By the the associative law of composition.
Identity: The identity element.
Inverse: Elements in the subset are all invertible.
4
5
No. A counter example is in :
but
.
6
((ab)c)d, (a(bc))d, a((bc)d), a(b(cd)
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(ab)c = ac = a,
a(bc) = ab = a.
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9
10
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Associativity:
Identity:
Inverse:
Closure: Inherited from G.
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