Algebra: Non-isomorphic groups

In summary: There are, in fact, two non-isomorphic groups of order two: the cyclic group of order two and the trivial group.In summary, there are two non-isomorphic groups of two elements: the cyclic group of order two and the trivial group. It is not possible to generate any other non-isomorphic groups of two elements.
  • #1
nuuskur
Science Advisor
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Homework Statement


How many non-isomorphic groups of two elements are there?

Homework Equations

The Attempt at a Solution


I don't understand exactly what we are being asked.
If we have a group of two elements under, say, addition, then [itex]G =\{0, g\}[/itex].
Then also [itex]g+g = 0[/itex] must be true, means [itex]g[/itex] is its own opposite. (of order 2).

Now, how should I construct a group under some operation ##*##: [itex]G' = \{e, g'\}[/itex], where ##e## is the unit/zero element (depending on operation) such that [itex]G[/itex] and [itex]G'[/itex] are not isomorphic?

[itex]G\cong G'[/itex] iff there exists a bijective group homomorphism [itex]f: G\to G'[/itex]
I can define [itex]f[/itex] such that:
[itex]f(0) = e[/itex] (satisfies one of the group homormorphism requirement)
[itex]f(g) = g'[/itex]
Is bijection and

[itex]f(0+g) = f(g) = g' = e*g'[/itex]
[itex]f(g+g) = f(0) = e = g'*g'[/itex]
so [itex]G\cong G'[/itex]

What must I do to generate non-isomorphic groups of two elements?

In general, for any ##n##, how can I determine the number of non-isomorphic groups of ##n## elements?
 
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  • #2
nuuskur said:
What must I do to generate non-isomorphic groups of two elements?
Do you think that is possible?

nuuskur said:
In general, for any ##n##, how can I determine the number of non-isomorphic groups of ##n## elements?
I think that is not easy at all.
 
  • #3
nuuskur said:

Homework Statement


How many non-isomorphic groups of two elements are there?

Homework Equations

The Attempt at a Solution


I don't understand exactly what we are being asked.
If we have a group of two elements under, say, addition, then [itex]G =\{0, g\}[/itex].
Then also [itex]g+g = 0[/itex] must be true, means [itex]g[/itex] is its own opposite. (of order 2).

Now, how should I construct a group under some operation ##*##: [itex]G' = \{e, g'\}[/itex], where ##e## is the unit/zero element (depending on operation) such that [itex]G[/itex] and [itex]G'[/itex] are not isomorphic?

[itex]G\cong G'[/itex] iff there exists a bijective group homomorphism [itex]f: G\to G'[/itex]
I can define [itex]f[/itex] such that:
[itex]f(0) = e[/itex] (satisfies one of the group homormorphism requirement)
[itex]f(g) = g'[/itex]
Is bijection and

[itex]f(0+g) = f(g) = g' = e*g'[/itex]
[itex]f(g+g) = f(0) = e = g'*g'[/itex]
so [itex]G\cong G'[/itex]

What must I do to generate non-isomorphic groups of two elements?

You can't. You are asked "how many non-isomorphic groups of two elements are there?" and you've shown that the answer is "all groups of order two are ismorphic to the cyclic group of order two."
 
  • #4
EDIT: nope
 

Related to Algebra: Non-isomorphic groups

1. What are non-isomorphic groups in algebra?

Non-isomorphic groups in algebra refer to groups that are not structurally equivalent. This means that while they may have the same number of elements and the same operations, they have different structures and cannot be transformed into one another through a change of variables.

2. How do you determine if two groups are non-isomorphic?

To determine if two groups are non-isomorphic, you can compare their structures. This involves examining the order of elements, the commutativity of operations, and the presence of subgroups. If any of these characteristics are different between the two groups, then they are non-isomorphic.

3. Why is it important to study non-isomorphic groups?

Studying non-isomorphic groups allows us to understand the complexities and variations within algebraic structures. It also helps us to develop a deeper understanding of the properties and behaviors of different groups, which can have practical applications in fields such as cryptography and coding theory.

4. Can a non-isomorphic group have the same number of elements as an isomorphic group?

Yes, it is possible for a non-isomorphic group to have the same number of elements as an isomorphic group. As mentioned before, the number of elements alone does not determine the structure of a group. It is the combination of elements and operations that make a group isomorphic or non-isomorphic.

5. What is an example of two non-isomorphic groups?

One example of two non-isomorphic groups is the cyclic group and the dihedral group. Both have the same number of elements, but their structures are different. The cyclic group is commutative, while the dihedral group is non-commutative. This difference in structure makes them non-isomorphic.

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