If cyclic group G is generated by an element  a of order n, then a\power{m} is a generator of G iff  (m,m)=1 i.e., the greatest common divisor  of m and n is 1

Q1// Let Ai be even permutation in Sn. Then * O(An)=n!/2 إضافة ملف Q2// Let Hi family of subgroups of (G, *). Prove that the intersection of Hi is also *.subgroup إضافة ملف

Let G be the cyclic group Z_4 under addition, of integers modulo 4. Let H be the subgroup of multiples of 2, {0,2}. Write out the two distinct cosets of H. One is 0+H=H. The other is some g+H. Then write out the addition table for the quotient group G/H consisting of those two cosets. The sum of a+H and b+H will be a+b+H.