How do you compose these functions? Let f: {1, 2, 3, 4} rightarrow {1, 2, 3, 4}

Question

How do you compose these functions?

Let f: {1, 2, 3, 4} rightarrow {1, 2, 3, 4} defined by f (1) = 3, f (3) = 2, f (2) = 4, and f (4) = 1. Then f is represented by the symbols (1324), (4132), (2413), and (3241). Similarly, we let the symbol (xyz) denote the function defined by f (x) = y, f (y) = z, f (z) = x, and f (w) = w. We let the symbol (xy) denote the function f (x) = y, f (y) = x, f (z) = z, and f (w) = w. Finally, the identity is denoted by 1. The 24 elements of S_4 are then: Find the group elements in the list above which are equal to the following compositions: (34) compositefunction (12) (123) compositefunction (23) (243) compositefunction (132)

Explanation / Answer

Consider (34)(12). Let us construct it's underlying function(s). Following the given example, (34) corresponds to the function
f(1) = 1., f(2) = 2, f(3) = 4 and f(4) =3.
Similarly (1,2) corresponds to the function; g(1) = 2, g(2) =1, g(3) = 3, g(4) =4.
Now (34)(12) = f(g). Let h = f(g). Then
h(1) = f(g(1) = f(2) = 2
h(2) = f(g(2)) = f(1) =1
h(3) = f(g(3)) = f(3) =4
h(4) = f(g(4)) =f(4) =3.
So h(1)=2, h(2) =1, h(3) =4, h(4) =3. Thus h corresponds to (12)(34);
So (34)(12) = (12)(34)

(123)(23)
Let f corresponds to (123) and g corresponds to (23).
Then f(1) =2, f(2) =3, f(3) =1, f(4) =4 and g(1) = 1, g(2) =3, g(3) =2, g(4) =4.
Let us compute h=f(g).
h(1) = f(g(1)) = f(1) = 2
h(2) = f(g(2)) =f(3) = 1
h(3) = f(g(3)) = g(2) = 3
h(4) = f(g(4)) = f(4) = 4.
So h(1) =2, h(2) =1, h(3) = 3, and h(4) =4.
Thus h corresponds to (12). Hence (123)(23)= (12)

(243)(132)

Let f correspond to (243) and g correspond to (132).
Then f(1) =1, f(2) =4, f(4) =3, f(3) =2 and
g(1) = 3, g(2) =1, g(3) =2, g(4) =4
Let h = f(g)
Then,
h(1) = f(g(1)) = f(3) =2
h(2) = f(g(2)) = f(1) =1
h(3) = f(g(3)) = f(2) = 4
h(4) = f(g(4)) = f(4) = 3.
Thus the element corresponding to h is (12)(34).
So (243)(132)= (12)(34).

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