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Group :- Let G be a non- empty set and ‘*’ be a binary
operation on G
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[ Note: * is a binary operation hence it must be
closed]
If a, b, c ∈ G then a*(b*c) = (a*b)*c
which represents that G show associative property
...

For each element a ∈ G there exist a’ of G such that
a*a’ = a’*a = e
which represents the existence of inverse
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Semi Group : Let G be a non-empty set and ‘*’ be a binary
operation on G
...

Associativity, if a, b, c 𝜖 G then a*(b*c) = (a*b)*c

SOME EAXMPLES
1
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Proof :- Z is a set of integers, we can write it as
Z = (−∞……………
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a+(b+c) = (a+b)+c {This shows
associative property}

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∃ 0 𝜖 Z such that a+0 = 0+a = a
{ This shows Existence of identity]
where 0 is an identity element for the
set of integers under addition operation
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a+b = b+a ∀ a, b 𝜖 Z
{This shows commutative property}

Since, it follows all five conditions of Abelian Group
Hence, (Z,+) is an Abelian Group
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Prove that (R-{0},X) is Abelian
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aX(bXc) = (aXb)Xc {This shows the
associative property}
∃ 1 𝜖 R-{0} such that
1Xa = aX1 = a
where 1 is an identity element for the
given set under the given operation,
which shows the Existence of Identity
element
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aXb = bXa {This shows the commutative
property}

Since, the given set R-{0} under the multiplication
operation follows all the five conditions of Abelian Group
Hence, (R-{0},X) is an Abelian Group

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