Solution :
(a)
(i)
Suppose that the vertices of the equilateral triangle be labelled 1,2,3. In other words, let X={1,2,3}.
Define the action . : G x X X given by, e.n = n for all n=1,2,3 and a.1 = 1, a.2 = 3, a.3 = 2. In other words, the action of the group element a on X is to reflect the equilateral triangle about the median passing through 1.
(ii)
Let the vertices of the square be labelled 1,2,3,4.
Equivalently, let X = {1,2,3,4}
Define the action . : G x X X by,
(0,0).n = n for all n=1,2,3,4
(1,0).1 = 2, (1,0).2 = 1, (1,0).3 = 4, (1,0).4 = 3
(0,1).1 = 3, (0,1).3 = 1, (0,1).2 = 4, (0,1).4 = 2
(1,1).1 = 4, (1,1).2 = 3, (1,1).3 = 2, (1,1).4 = 1
In other words, the action of (1,0) is to reflect the square along the central line perpendicular to the segments [1 2] and [3 4]. The action of (0,1) is to reflect the square along the central line perpendicular to the segments [1 3] and [2 4]. The action of (1,1) is the composite of the previous two actions.
(b)
The fact that the two maps defined above are indeed group actions of G on X is evident.
Firstly, by definition, in each case, e.a = a for all a in X. Secondly, g1.(g2.a) = (g1g2).a for all g1,g2 in G and a in X.
Hence, . is indeed a group action in the above two cases.
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