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Index of subgroup and coset of subgroup

Explain what the index of a subgroup and a coset of a group are. Also, prove that if N is a subgroup of a group G such that [G: N] = 2, and if "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.

Solution Preview

If N is a subgroup of G, the notation [G: N] = 2 means that 2 is the index of N in G, which means that there are two cosets of N in G. Specifically, there are 2 right cosets of N in G (namely, N and N*a for some element "a" which is not an element of N), and there are two left cosets of N in G (namely, N and a*N).

If N is a subgroup of G, then |N| * [G: N] = |G| (where, for any set S, |S| denotes the cardinality of S). From this it follows that if G is finite, then the order of every subgroup of G is some divisor of |G|.

The cosets form a partition of G: Every element of G is an element of one and only one left coset of G, and of one and only one right coset of G. Thus every pair of right cosets are either identical (to each other) or disjoint, and similarly for every pair of left cosets. Moreover, all the cosets are of equal cardinality.

Thus if [G: N] = 2, then the number of elements of G that are in N is equal to the number of elements of G that are not in N. Note that if "a" is an element of G which is not in N, then the left coset a*N ...

Solution Summary

The meanings of index of a subgroup and coset of a subgroup are explained. A detailed proof of the following statement is given: Let N be a subgroup of a group G such that [G: N] = 2. If "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.

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