Abstract Algebra Answers

Suppose that G is a group of order 60. What are the possible orders of a subgroup
H ≤ G?

Assume that G is a finite group, and that there exist g,h are element of G, g doesnt equal h, such that order(g)=p, order(h)=q, where p and q are distinct primes. What can be concluded about |G|?

Assume that G is a group, H is a subgroup of G, |H|=6, [G:H]>4 and |G|<50. What can be concluded about |G|?

Assume that (G,*) is a group, and that |G|=p, where p is prime. Prove that G is cyclic.

Express the following as a product of disjoint cycle. Check them for being even or odd permutation and find the inverse of each of them in S7
1. (1 4 7) (2 6 5) (2 4 1) (5 6 7)
2. (7 1) (2 6 5) (2 4 1) (7 5 6)

Express the following permutation in S7 in two line notation
(1 4 7) ( 2 6 5)

If R is noetherian and I is a ideal show that R/I is noetherian.

Direct sum of neotherian module's is neotherian

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