Groups of Order 144: Abelian Groups Up to Isomorphism

In summary, there are 10 non-isomorphic groups of order 144, with the missing isomorphism being Z16 x Z9. This information was provided in response to a question asking for a list of all abelian groups of order 144. The original poster had only listed 9 groups and was unsure of the missing one, but later realized it was Z16 x Z9. The conversation took place during finals, causing some confusion for the original poster.
  • #1
BurrowK
3
0
I'm going insane. The question is:
List all abelian groups (up to isomorphism) of order 144.

There are 10 non-isomorphic groups of order 144 and I only have 9. Here they are:

Z2 X Z2 X Z2 X Z2 X Z3 X Z3
Z2 X Z2 X Z2 X Z2 X Z9
Z4 X Z2 X Z2 X Z3 X Z3
Z4 X Z2 X Z2 X Z9
Z8 X Z2 X Z3 X Z3
Z8 X Z2 X Z9
Z4 X Z4 X Z3 X Z3
Z4 X Z4 X Z9
Z16 X Z3 X Z3

(with the isomorphism being Z16 X Z9)

Anyone? Bueller?
 
Last edited:
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  • #2
BurrowK said:
(with the isomorphism being Z16 X Z9)
What do you mean by this statement? Z16 x Z9 gives a distinct isomorphism class - it's precisely the one you're missing.
 
  • #3
Yeah, I realized that that counted towards the 10. It's the midst of finals so the brain's on the fritz. I was just going to take this post down, thanks though!
 

FAQ: Groups of Order 144: Abelian Groups Up to Isomorphism

What is a group of order 144?

A group of order 144 is a mathematical structure consisting of 144 elements that follow a set of rules called group axioms. These axioms include closure, associativity, identity, and inverse elements.

What is an abelian group?

An abelian group is a group in which the order of operations does not affect the outcome. This means that the elements of the group commute with each other, resulting in a commutative group operation.

How many abelian groups of order 144 exist?

There are 5 abelian groups of order 144, up to isomorphism. This means that there are 5 distinct structures that can be formed with 144 elements that follow the group axioms.

What are the 5 abelian groups of order 144?

The 5 abelian groups of order 144 are: Z2 x Z2 x Z2 x Z2 x Z3 x Z3, Z2 x Z2 x Z2 x Z3 x Z4, Z2 x Z2 x Z6 x Z6, Z2 x Z2 x Z2 x Z8 x Z9, and Z2 x Z2 x Z4 x Z9 x Z9.

How are these abelian groups of order 144 different from non-abelian groups of the same order?

The main difference between abelian groups and non-abelian groups of the same order is the commutativity property. In abelian groups, the order of operations does not affect the outcome, while in non-abelian groups, the order of operations can change the result. Additionally, abelian groups have a simpler structure compared to non-abelian groups.

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