Group Theory

Forum for the GRE subject test in mathematics.
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Isosceles
Posts: 1
Joined: Thu Dec 19, 2024 6:37 am

Group Theory

Post by Isosceles » Fri Jan 17, 2025 1:27 am

Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G?

poly
Posts: 16
Joined: Sun Aug 04, 2024 6:59 pm

Re: Group Theory

Post by poly » Fri Jan 17, 2025 2:37 am

Math stackexchange is where you ask actual math questions I think.

For this: Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. The orders of these cyclic groups must divide 16. Further, 4x=0 meaning the generator has order at most 4. So, they are just direct products of Z2 and Z4s with order 16.

fredholm.alternative
Posts: 20
Joined: Mon Oct 07, 2024 9:04 pm

Re: Group Theory

Post by fredholm.alternative » Fri Jan 17, 2025 3:08 pm

poly wrote:
Fri Jan 17, 2025 2:37 am
Math stackexchange is where you ask actual math questions I think.

For this: Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. The orders of these cyclic groups must divide 16. Further, 4x=0 meaning the generator has order at most 4. So, they are just direct products of Z2 and Z4s with order 16.
ironically i feel like i've seen this exact question in a practice math gre, the namesake of this forum.



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