1
Subgroups
Notation: When it’s obvious that the group operation is addition (for example
when 𝐺 = ℤ) we may write 𝑎 + 𝑏 instead of 𝑎 ∗ 𝑏. Otherwise, we’ll write 𝑎𝑏
instead of 𝑎 ∗ 𝑏.
We will also write:
𝑎𝑛 = (𝑎)(𝑎)(𝑎) … (𝑎) 𝑛 times
𝑎−1 = inverse of 𝑎
𝑎−𝑛 = (𝑎 −1 )(𝑎−1 ) … (𝑎−1 ) 𝑛 times
𝑎0 = 𝑒.
Notice that 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 ; 𝑚, 𝑛 ∈ ℤ.
Ex. 𝑎−2 𝑎4 = (𝑎−1 )(𝑎−1 )(𝑎)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑎−1 𝑎)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑒)(𝑎)(𝑎)(𝑎)
= (𝑎−1 𝑒)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑎)(𝑎)(𝑎)
= (𝑎−1 𝑎)(𝑎)(𝑎)
= 𝑒(𝑎)(𝑎)
= 𝑎2 .
Def. If 𝐺 is a group, then the order of 𝐺, written |𝐺|, is the number of
elements in 𝐺.
, 2
Def. If a subset 𝐻 of a group 𝐺 is closed under the binary operation of 𝐺 and if 𝐻
is a group with that binary operation, then 𝐻 is a subgroup of 𝐺. We will write
𝐻 ≤ 𝐺 or 𝐺 ≥ 𝐻 in that case.
𝐻 < 𝐺 or 𝐺 > 𝐻 will mean 𝐻 ≤ 𝐺 but 𝐻 ≠ 𝐺
Ex. (ℤ, +) ≤ (ℝ, +), in fact (ℤ, +) < (ℝ, +),
since ℤ ⊊ ℝ and ℤ and ℝ are both groups under +.
Ex. (ℚ+ , +) is not a subgroup of (ℝ, +) even though ℚ+ ⊆ ℝ.
This is because ℚ+ is a group under ∙ not + (under +, ℚ+ doesn’t
contain inverses for all of its elements).
Def. If 𝐺 is a group, then the subgroup consisting of 𝐺 itself is called the
improper subgroup of 𝑮. All the other subgroups are proper subgroups.
The subgroup {𝑒} is called the trivial subgroup of 𝑮. All other subgroups are
called nontrivial.
Ex. Let 𝐺 = ℝ𝑛 with vector addition as the binary operation. This is a group
under +. Let 𝐻 be the set of vectors in ℝ𝑛 having 0 as the entry in the first
component. Show 𝐻 is a subgroup of 𝐺.
0) 𝐻 is closed under +:
< 0, 𝑎2 , 𝑎3 , … , 𝑎𝑛 > +< 0, 𝑏2 , 𝑏3 , … , 𝑏𝑛 >
= < 0, 𝑎2 + 𝑏2 , … , 𝑎𝑛 + 𝑏𝑛 > ∈ 𝐻.
1) + is associative on 𝐻 because vector addition is associative.
Subgroups
Notation: When it’s obvious that the group operation is addition (for example
when 𝐺 = ℤ) we may write 𝑎 + 𝑏 instead of 𝑎 ∗ 𝑏. Otherwise, we’ll write 𝑎𝑏
instead of 𝑎 ∗ 𝑏.
We will also write:
𝑎𝑛 = (𝑎)(𝑎)(𝑎) … (𝑎) 𝑛 times
𝑎−1 = inverse of 𝑎
𝑎−𝑛 = (𝑎 −1 )(𝑎−1 ) … (𝑎−1 ) 𝑛 times
𝑎0 = 𝑒.
Notice that 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 ; 𝑚, 𝑛 ∈ ℤ.
Ex. 𝑎−2 𝑎4 = (𝑎−1 )(𝑎−1 )(𝑎)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑎−1 𝑎)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑒)(𝑎)(𝑎)(𝑎)
= (𝑎−1 𝑒)(𝑎)(𝑎)(𝑎)
= (𝑎−1 )(𝑎)(𝑎)(𝑎)
= (𝑎−1 𝑎)(𝑎)(𝑎)
= 𝑒(𝑎)(𝑎)
= 𝑎2 .
Def. If 𝐺 is a group, then the order of 𝐺, written |𝐺|, is the number of
elements in 𝐺.
, 2
Def. If a subset 𝐻 of a group 𝐺 is closed under the binary operation of 𝐺 and if 𝐻
is a group with that binary operation, then 𝐻 is a subgroup of 𝐺. We will write
𝐻 ≤ 𝐺 or 𝐺 ≥ 𝐻 in that case.
𝐻 < 𝐺 or 𝐺 > 𝐻 will mean 𝐻 ≤ 𝐺 but 𝐻 ≠ 𝐺
Ex. (ℤ, +) ≤ (ℝ, +), in fact (ℤ, +) < (ℝ, +),
since ℤ ⊊ ℝ and ℤ and ℝ are both groups under +.
Ex. (ℚ+ , +) is not a subgroup of (ℝ, +) even though ℚ+ ⊆ ℝ.
This is because ℚ+ is a group under ∙ not + (under +, ℚ+ doesn’t
contain inverses for all of its elements).
Def. If 𝐺 is a group, then the subgroup consisting of 𝐺 itself is called the
improper subgroup of 𝑮. All the other subgroups are proper subgroups.
The subgroup {𝑒} is called the trivial subgroup of 𝑮. All other subgroups are
called nontrivial.
Ex. Let 𝐺 = ℝ𝑛 with vector addition as the binary operation. This is a group
under +. Let 𝐻 be the set of vectors in ℝ𝑛 having 0 as the entry in the first
component. Show 𝐻 is a subgroup of 𝐺.
0) 𝐻 is closed under +:
< 0, 𝑎2 , 𝑎3 , … , 𝑎𝑛 > +< 0, 𝑏2 , 𝑏3 , … , 𝑏𝑛 >
= < 0, 𝑎2 + 𝑏2 , … , 𝑎𝑛 + 𝑏𝑛 > ∈ 𝐻.
1) + is associative on 𝐻 because vector addition is associative.
