Prove that if a finite group G has a unique subgroup of order m for each...

Prove that if a finite group G has a unique subgroup of order m for each divisor m of the order n of G then G is cyclic.

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Colby Messinger answered 1 year ago

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

(Modern Algebra) Show that if G is a finite group that has at most one subgroup...

(Modern Algebra) Show that if G is a finite group that has at most one subgroup for each divisor of its order then G is cyclical.

show that if H is a p sylow subgroup of a finite group G then for...

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with f|U = IdU

If G has no nontrivial subgroups, prove that G must be finite of prime order.

abstract algebra Let G be a finite abelian group of order n Prove that if d...

abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.

In a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size

Prove that in a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size.

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Prove that if a finite group G has a unique subgroup of order m for each...

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