Math, asked by atharv4777, 1 year ago

Prove that a finite abelian group is cyclic if and only if all of its sylow sub group is cyclic

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Answered by swapnil16
0
Assume that G is a finite abelian group and is simple, then G has no nontrivial normal subgroups. (Now I don't know how to show that this implies that G has order p, where p is prime.

(<=)Assume that G is a finite abelian group with order p, where p is a prime. (Since the order of p is prime then what does this mean?)

Edit: Can someone check my new attempt at the proof?

(=>) Suppose G is a simple finite abelian group. Suppose for the sake of contradiction that G does not have prime order, then |G|=p*k where p is a prime number and k is an integer such that k>1. Then G has an element of order p. Let the element of order p be called x. Then , the subgroup generated by x, is of order p and is not all of G. Since G is abelian, this subgroup is normal, which leads us to a contradiction. Therefore, G must have prime order.

(<=) Suppose that G is a finite abelian group and it’s order is p, a prime. Since G has prime order, then the only two subgroups of G are the trivial subgroup and the group G. Then, by definition the group G is simple since there are no nontrivial proper subgroups, and thus no nontrivial normal subgroups.

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