Question

What is an abelian group?

Answer 1

Answer:

Abelian groups are generally simpler to analyze than nonabelian groups are, as many objects of interest for a given group simplify to special cases when the group is abelian. For example, the conjugacy classes of an abelian group consist of singleton sets (sets containing one element), and every subgroup of an abelian group is normal.

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

Answer :

Abelian group :

If a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

ie . if x*y = y*x ∀ x , y ∈ (G,*) , then the group G is said to be abelian .

Examples :

• (Z , +) is an abelian group , ie. the set of integers is a group with respect to addition .
• (Zₙ , +ₙ) is an abelian group , ie. Zₙ is a group with respect to addition modulo n .
• (R , •) is an abelian group , ie. the set of real numbers is a group with respect to multiplication .