Group Theory
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Group (Defn)
A non-empty set G with binary operation • or ∗ is said to be group if G satisfies groups laws, 〈group denoted by (G,∗)〉
Groups laws
1.Closure
law
a,b ϵ G ⇨ a ∗ b ϵ G
2.Associative law
a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)
3.The existence of Identity
the exists e ϵ G such that
a ∗ e = e ∗a = a, ⩝ a ϵ G
4.The existence of Inverse
there exists aʹϵ G such that aʹ∗a
= a∗aʹ, ⩝ a ϵ G.
Commutative Group or Abelian Group
Let G be a group with binary operation ∗ 〈i.e (G,∗) be a group〉
then for every a,b ϵ G such that a∗b =b∗a
Quasi Group
Let G be a non
empty set then G satisfies closure law only
a,b ϵ G ⇨ a ∗ b ϵ G
Semi-Group
Let
G be a non empty set then G satisfies closure law and
associative
law only
closure: a,b ϵ G ⇨ a ∗ b ϵ G
associative:
a,b,c ϵ
G ⇨(a∗b)∗c =a∗(b∗c)
Monoid
Let
G be a non empty set then G satisfies closure law ,
associative law and identity existence
only
closure : a,b ϵ G ⇨ a ∗ b ϵ G
associative : a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)
identity : the exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G
Hints:
|
Closure |
Associative |
Identity |
Inverse |
Commutative |
Quasi |
Yes |
No |
No |
No |
No |
Semi-group |
Yes |
Yes |
No |
No |
No |
Monoid |
Yes |
Yes |
Yes |
No |
No |
Group |
Yes |
Yes |
Yes |
Yes |
No |
Abelian group |
Yes |
Yes |
Yes |
Yes |
Yes |
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