1
Groups
Def. A group (𝐺,∗) is a set 𝐺, and a binary operation ∗, such that the following
axioms hold:
0) 𝐺 is closed under ∗
1) For all 𝑎, 𝑏, 𝑐 ∈ 𝐺 we have
(𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐) i.e. ∗ is associative
2) There is an element 𝑒 ∈ 𝐺 such that
for all 𝑥 ∈ 𝐺, 𝑒 ∗ 𝑥 = 𝑥 ∗ 𝑒 = 𝑥.
𝑒 is called the identity element.
3) To each 𝑎 ∈ 𝐺 there exists an element 𝑎′ ∈ 𝐺
such that 𝑎 ∗ 𝑎′ = 𝑎′ ∗ 𝑎 = 𝑒.
𝑎′ is called the inverse of 𝑎.
Def. A group 𝐺 is abelian if its binary operation is commutative.
Ex. Show that (ℤ, +) is a group (so are (ℚ, +), (ℝ, +), and (ℂ, +)).
0) ℤ is closed under + .
1) Addition in ℤ is associative.
2) 0 ∈ ℤ is the identity element.
3) For any 𝑎 ∈ ℤ, −𝑎 ∈ ℤ is the inverse of 𝑎.
(ℤ, +) is also an abelian group because + is commutative.
, 2
Ex. Show that (ℤ+ , +) is not a group.
0) ℤ+ is closed under +.
1) + is associative.
2) There is no identity element (0 ∉ ℤ+ ).
3) No element of ℤ+ has an inverse (−𝑎 ∉ ℤ+ ) in ℤ+ .
So (ℤ+ , +) fails axioms 2 and 3.
Ex. ℚ+ , ℝ+ , ℚ∗ , ℝ∗ and ℂ∗ are all abelian groups under multiplication.
0) Each set is closed under multiplication.
1) Multiplication is associative (and commutative).
2) 1 is the identity element.
1
3) If 𝑎 is in any of the above sets, so is , the multiplicative inverse.
𝑎
Ex. Show the set 𝐹 of all real valued functions on ℝ is an abelian group under
addition.
0) 𝐹 is closed under addition.
1) Addition of functions is associative (and commutative).
2) 𝑓(𝑥 ) = 0 is the identity element.
3) If 𝑓 (𝑥 ) ∈ 𝐹 then −𝑓(𝑥 ) ∈ 𝐹 and −𝑓(𝑥 ) is the inverse of 𝑓 (𝑥 ).
Groups
Def. A group (𝐺,∗) is a set 𝐺, and a binary operation ∗, such that the following
axioms hold:
0) 𝐺 is closed under ∗
1) For all 𝑎, 𝑏, 𝑐 ∈ 𝐺 we have
(𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐) i.e. ∗ is associative
2) There is an element 𝑒 ∈ 𝐺 such that
for all 𝑥 ∈ 𝐺, 𝑒 ∗ 𝑥 = 𝑥 ∗ 𝑒 = 𝑥.
𝑒 is called the identity element.
3) To each 𝑎 ∈ 𝐺 there exists an element 𝑎′ ∈ 𝐺
such that 𝑎 ∗ 𝑎′ = 𝑎′ ∗ 𝑎 = 𝑒.
𝑎′ is called the inverse of 𝑎.
Def. A group 𝐺 is abelian if its binary operation is commutative.
Ex. Show that (ℤ, +) is a group (so are (ℚ, +), (ℝ, +), and (ℂ, +)).
0) ℤ is closed under + .
1) Addition in ℤ is associative.
2) 0 ∈ ℤ is the identity element.
3) For any 𝑎 ∈ ℤ, −𝑎 ∈ ℤ is the inverse of 𝑎.
(ℤ, +) is also an abelian group because + is commutative.
, 2
Ex. Show that (ℤ+ , +) is not a group.
0) ℤ+ is closed under +.
1) + is associative.
2) There is no identity element (0 ∉ ℤ+ ).
3) No element of ℤ+ has an inverse (−𝑎 ∉ ℤ+ ) in ℤ+ .
So (ℤ+ , +) fails axioms 2 and 3.
Ex. ℚ+ , ℝ+ , ℚ∗ , ℝ∗ and ℂ∗ are all abelian groups under multiplication.
0) Each set is closed under multiplication.
1) Multiplication is associative (and commutative).
2) 1 is the identity element.
1
3) If 𝑎 is in any of the above sets, so is , the multiplicative inverse.
𝑎
Ex. Show the set 𝐹 of all real valued functions on ℝ is an abelian group under
addition.
0) 𝐹 is closed under addition.
1) Addition of functions is associative (and commutative).
2) 𝑓(𝑥 ) = 0 is the identity element.
3) If 𝑓 (𝑥 ) ∈ 𝐹 then −𝑓(𝑥 ) ∈ 𝐹 and −𝑓(𝑥 ) is the inverse of 𝑓 (𝑥 ).
