Abelian Group and “Group”

I am having trouble distinguishing some terms here. I got these from my book

A group is a nonempty set G together with binary operation * that satisfies the three properties

G1. Associativity

G2. Identity

G3. Inverse

If a group also satisfies Commutativity, then it is abelian, if it doesn't then it is nonabelian

Okay here is my confusion, the last statement seems to throw the term group aside. Is it basically telling me to check whether Commutativity is satisfied or not and label them abelian and nonabelian?

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But what about "If a group"...? –  draks ... Oct 10 '12 at 5:45

5 Answers

A group $\{G,*\}$ satisfies the following conditions:

1. $a*b\in G, \forall a,b\in G$

2. $(a*b)*c=a*(b*c), \forall a,b,c\in G$

3. $\exists! e\in G:e*a=a*e=a,\forall a\in G$

4. $\exists! a^{-1}:a a^{-1}=a^{-1}a=e, \forall a\in G$

An abelian group is just a specific type of group which also satisfies $$a*b=b*a,\ \ \forall a,b\in G$$ If it does not meet this condition it is non-abelian.

A useful analogy might be to consider the natural numbers. A number is prime if it has no positive divisors other than 1 and itself. If it does not meet this condition it is non-prime (also known as composite). Similarly, a group is abelian if all its elements commute, otherwise it is non-abelian.

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Abelianness (commutativity) is a property a group can have. If G is a group and * also is commutative, then G "is abelian". We can call G an "abelian group" or just say "G is abelian". The same goes for nonabelian.

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You seem to be getting confused over two types of adjective usage. Wikipedia highlights the difference with this example:

• That's an interesting idea. (attributive)
• That idea is interesting. (predicative)

So, it's legitimate to say e.g.:

• $\mathbb{Z}_n$ is an abelian group.
• $\mathbb{Z}_n$ is abelian.

And they have the same meaning.

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The adjective abelian typically only applies to groups. If a ring satisfies the commutativity condition, then we call the ring commutative (or noncommutative if it doesn't) rather than abelian or nonabelian. So if you see a structure described simply as abelian, it's probably understood to be a group. With regards to your specific question, when referring to a group $G$ satisfying the commutativity property, calling $G$ abelian or calling $G$ an abelian group are both acceptable.

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Not every group is commutative. For instance, check the group of $2\times 2$ matrices under multiplication with determinant does not equal to zero. This group in not abelian (commutative) group since $AB \neq BA$ in general.

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