Question

Prove that any group of order 9 is abelian.

Answer #1

**Solution
:**

LetG be a group (not necessarily an Abelian group) of order 425.
Prove that G must have an element of order 5.

If n is a square-free integer, prove that an abelian group of
order n is cyclic.

Let G be a group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

A)
Prove that a group G is abelian iff (ab)^2=a^2b^2 fir any two
ekemwnts a abd b in G.
B) Provide an example of a finite abelian group.
C) Provide an example of an infinite non-abelian group.

Let G be a group of order p^3. Prove that either G is abelian or
its center has exactly p elements.

Abstract Algebra (Modern Algebra)
Prove that every subgroup of an abelian group is abelian.

Characterize those integers n such that any Abelian group of
order n belongs to one of exactly four isomorphism classes.

Characterize those integers n such that any abelian group of
order n belongs to one of exactly two isomorphism classes.

Prove that if G is a group with |G|≤5 then G is abelian.

Show that a group of order 5 is abelian.

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