Group Theory
Group Theory
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?
Re: Group Theory
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.
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.
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Re: Group Theory
ironically i feel like i've seen this exact question in a practice math gre, the namesake of this forum.poly wrote: ↑Fri Jan 17, 2025 2:37 amMath 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.