p 2 is even.Ĭase 1: Given that p1 and p2 are two permutations of the same set or two different sets such that both of them are even, then p1 and p2 are both results of 2a and 2b transpositions, respectively. ![]() In contrast, a permutation formed by using an even number of permutations is known as an even permutation.Ī look at the properties of even and odd permutations:ġ) Given that p 1 and p 2 are two permutations of different sets or the same set such that both p 1 and p 2 are either even or odd, then the product of these permutations p 1. Even and Odd PermutationsĪ permutation which can be formed using an odd number of transpositions is called an odd permutation. In A 5 goes to 5 and in B 5 goes to 1 so in A.B 5 goes to 1 and finally in A 1 goes to 1 and in B 1 goes to 4 so in A.B 1 goes to 3. In A 4 goes to 2 and in B 2 goes to 2 so in A.B 4 goes to 2. ![]() Similarly, in A 3 goes to 4 and in B 4 goes to 5 so in A.B 2 goes to 5. We can see that in the permutation A 2 goes to 3 and in B 4 goes to 4 thus, in A.B 2 goes to 4. Let us try to look into what is happening here. Let us assume that a set B has the following elements A permutation of a set is the one-one onto mapping of the set to itself, that is, bijective mapping. If there are n numbers in the original set, then its cardinal number is also n, and the number of elements in its permutation group equals n! Definition of Permutation GroupĪ permutation group of a set is another set which contains all possible permutations that can be created from the original set. The permutation group of a set can be defined as a set containing all the possible permutations that can be created for the original set. If the cardinal number (the total number of elements in a set) of a finite set is n, then n is also called the degree of permutation of the set. A bijective relation refers to one-one mapping. ![]() A permutation of a finite set is a bijective relation from itself to itself. $S_3$ has 3 elements of order 2, none of which commute with the others.A permutation of a set is defined for finite sets only. Now you have a bit of work to do here, since there are several elements of order 2, and several of order 3 but by the inherent symmetries of the set-up, you can assume the element of order 3 is $(123)$, and then there aren't so many cases you have to look at.ĮDIT: but here's an easier way to handle $S_3$. ![]() Are there any in $A_4$?įor a copy of $S_3$, you need an element $a$ of order 2, and an element $b$ of order 3, and you need $ba=ab^2$. Are there such elements in $A_4$?įor a cyclic group of order 6, you need an element of order 6. Are there any in $A_4$?įor a Klein 4-group, you need three elements of order 2, each of which commutes with the other two. If you know that the only groups of order 4 (up to isomorphism) are the cyclic group and the Klein 4-group, and that the only groups of order 6 (up to isomorphism) are the cyclic group and $S_3$, then you can just look for copies of those groups in $A_4$.įor a cyclic group of order 4, you need an element of order 4.
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