✔ 最佳答案
1 Let X be a set. Let Sym(X) be the set of permutations of X (i.e. the set of bijective functions from X to itself). Then the act of taking the composition of two permutations induces a group structure on Sym(X) . We call this group the symmetric group. Cayley's Theorem means that if G be a group, then G is isomorphic to a subgroup of the permutation group S_G
2 g2∈g1H, then g2=g1h where h is an element of H
So for x ∈ g2H, its format should be g1hk where k is another element of H
Since H is a group, hk should become another element of H, called it t
Just x=g1t ∈g1H
Alternatively, assume that y ∈g1H, its format should be g1k where k is another element of H. Since g2=g1h and H is a group => g1=g2h^(-1). So y=g1k=g2h^(-1)k=g2t where t is another element of H. Just y ∈g2H
Combining the results, we have g1H = g2H
If there is an element z belongs to both g1H and g2H, then z=g1k=g2t where k and t are elements of H. From this we can deduce that g2=g1kt^(-1) and so g2 ∈g1H => g1H = g2H..This prove that any two left cosets of H are equal or disjoint.
