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🏠
/
Mathematics
/
Group Theory
/
Group
Group
A set equipped with a binary operation that satisfies closure, associativity, identity, and invertibility.
📜
The statement of the theorem
A group is a set
G
G
G
with an operation
⋅
\cdot
⋅
such that:\n1.
(
a
b
)
c
=
a
(
b
c
)
(ab)c = a(bc)
(
ab
)
c
=
a
(
b
c
)
\n2.
∃
e
∈
G
\exists e \in G
∃
e
∈
G
s.t.
a
e
=
e
a
=
a
ae = ea = a
a
e
=
e
a
=
a
\n3.
∀
a
∈
G
,
∃
b
∈
G
\forall a \in G, \exists b \in G
∀
a
∈
G
,
∃
b
∈
G
s.t.
a
b
=
b
a
=
e
ab = ba = e
ab
=
ba
=
e
.