Question

Example of abelian and non abelian group

Answer 1

In general, matrices, even invertible matrices, do not form an abelian groupunder multiplication because matrix multiplication is generally notcommutative. However, some groupsof matrices are abelian groups under matrix multiplication – one example is the group of 2×2 rotation matrices


The quaternion group is non-abelianand of order 8. ... You can also construnt non-abelian finite groupsfrom finite abelian groups. Forexample, any cyclic group will beabelian. However, the wreath product C n ≀ C n is an example of a non-abelianfinite group (well strictly speaking I should say for n ≥ 2 here).

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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 .

Non-abelian group :

If a group (G,*) doesn't hold commutative property , then it is called non-abelian group .

Examples :

• The set of n×n non-singular matrices is non-abelian with respect to matrix multiplication .
• Dihedral group D₂ₙ (the group of symmetries of a regular n-gon) is a non-abelian group .
• The quaternion group Q₈ is a non-abelian group .