Up to isomorphism

The previous post showed that there are 10 Abelian groups that have 2025 elements. Implicitly that means there are 10 Abelian groups up to isomorphism, i.e. groups that are not in some sense “the same” even if they look different.

Sometimes it is clear what we mean by “the same” and there’s no need to explicitly say “up to isomorphism” and doing so would be pedantic. Other times it helps to be more explicit.

In some context you want to distinguish isomorphic as different objects. This is fine, but it means that you have some notion of “different” that is more strict than “not isomorphic.” For example, the x-axis and the y-axis are different subsets of the plane, but they’re isomorphic as 1-dimensional vector spaces.

Abelian groups

There is a theorem that says ℤ_mn, the group of integers mod mn, is isomorphic to the direct sum ℤ_n ⊕ ℤ_n if and only if m and n are relatively prime. This means, for example, that ℤ₁₅ and ℤ₃ ⊕ ℤ₅ are isomorphic, but ℤ₉ and ℤ₃ ⊕ ℤ₃ are not.

Because of this theorem it’s possible to come up with a list of Abelian groups of order 2025 that looks different from the list in the previous post but it actually the same, where “same” means isomorphic.

In the previous post we listed direct sums of groups where each group was a cyclic group of some prime power order:

• ℤ₈₁ ⊕ ℤ₂₅
• ℤ₈₁ ⊕ ℤ₅ ⊕ ℤ₅
• ℤ₂₇ ⊕ ℤ₃ ⊕ ℤ₂₅
• ℤ₂₇ ⊕ ℤ₃ ⊕ ℤ₅ ⊕ ℤ₅
• ℤ₉ ⊕ ℤ₉ ⊕ ℤ₂₅
• ℤ₉ ⊕ ℤ₉ ⊕ ℤ₅ ⊕ ℤ₅
• ℤ₉ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₂₅
• ℤ₉ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₅ ⊕ ℤ₅
• ℤ₃ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₂₅
• ℤ₃ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₅ ⊕ ℤ₅

We could rewrite this list as follows by combining group factors of relatively prime orders:

• ℤ₂₀₂₅
• ℤ₅ ⊕ ℤ₄₀₅
• ℤ₃ ⊕ ℤ₆₇₅
• ℤ₁₅ ⊕ ℤ₁₃₅
• ℤ₉ ⊕ ℤ₂₂₅
• ℤ₄₅ ⊕ ℤ₄₅
• ℤ₃ ⊕ ℤ₃ ⊕ ℤ₂₉₅
• ℤ₃ ⊕ ℤ₁₅ ⊕ ℤ₄₅
• ℤ₃ ⊕ ℤ₃ ⊕ ℤ₃ ⊕ ℤ₇₅
• ℤ₃ ⊕ ℤ₃ ⊕ ℤ₁₅ ⊕ ℤ₁₅

This listing follows a different convention, namely that the order of each group is a factor of the order of the next.

Related posts

• NBA and MLB trees are isomorphic
• Field of order 9
• Number of representations of a finite field
• Duals and double duals
• Groups versus Abelian Groups

The post Up to isomorphism first appeared on John D. Cook.