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#1 The Definition of Group

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(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)

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(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

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Associativity: By the the associative law of composition.
Identity: The identity element.
Inverse: Elements in the subset are all invertible.

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No. A counter example is in :
but .

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((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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,

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,

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Associativity:
Identity:
Inverse:
Closure: Inherited from G.