Finite Groups|Subgroups
Angelica L. Jemong
Department of Mathematics and Statistics
College of Science and Mathematics
MSU-Iligan Institute of Technology
Isomorphic Binary Structures
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Finite Groups
• The set 𝑒 is the minimal set that will give rise to a group, where 𝑒 is the
identity element.
• Up to isomorphism, there is only one group of 2 elements which the following
defined binary operation:
Such group is ℤ2 .
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Finite Groups
• Up to isomorphism, there is only one group of 3 elements which the following
defined binary operation:
Such group is ℤ3 .
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Finite Groups
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Order of a Group
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Example 1
▪ Consider the group ℤ under addition. Then ℤ has infinite order.
▪ The group 𝑈 10 = {1,3,7,9} under multiplication modulo 10 has
order 4.
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Order of an Element
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Example 2
▪ Consider 𝑈 15 under multiplication modulo 15.
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Example 3
▪ Consider ℤ10 under addition modulo 10.
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Subgroup
Notations:
𝐻≤𝐺 means H is a subgroup of G
𝐻<𝐺 means H is a proper subgroup of G
(H is a subgroup of G, but 𝐻 ≠ 𝐺)
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Subgroup
Proof left as an exercise.
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Subgroup
Examples:
▪ 𝑒 is a trivial subgroup of G
▪ 𝐻 ≠ {𝑒} is nontrivial subgroup of G.
▪ Question: Is ℤ𝑛 under addition modulo n a subgroup of ℤ under
addition?
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Subgroup
Consider the groups of order 4.
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Subgroup
Consider the groups of order 4.
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Subgroup
Consider the groups of order 4.
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Subgroup Criterion
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Example 4
▪ Let G be an Abelian group with identity e.
Then 𝐻 = 𝑥 ∈ 𝐺 | 𝑥 2 = 𝑒 is a subgroup of G.
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Example 5
▪ Let G be an Abelian group and H and K be subgroups of G.
Then 𝐻𝐾 = ℎ𝑘| ℎ ∈ 𝐻, 𝑘 ∈ 𝐾 is a subgroup of G.
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Exercise:
▪ Let G be an Abelian group under multiplication with identity e.
Then 𝐻 = 𝑥 2 | 𝑥 ∈ 𝐺 is a subgroup of G.
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How do you prove that a subset of a group is not a subgroup?
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Example 6
Let 𝐺 be a group of nonzero real numbers under multiplication.
Consider the following sets:
𝐻 = 𝑥 ∈ 𝐺|𝑥 = 1 𝑜𝑟 𝑥 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙
𝐾 = 𝑥 ∈ 𝐺|𝑥 ≥ 1
Question: Is 𝐻 a subgroup of 𝐺?
Question: Is 𝐾 a subgroup of 𝐺?
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Cyclic Subgroups
For any element 𝑎 from a group, we let 𝑎 denote the set
𝑎𝑛 |𝑛 ∈ ℤ .
• Note that a0 is defined to be the identity.
• Although the list … , 𝑎−2 , 𝑎−1 , 𝑎0 , 𝑎1 , 𝑎2 ,… has infinitely many
entries, the set 𝑎 may have only finitely many elements.
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Cyclic Subgroups
Theorem.
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Cyclic Subgroups
The subgroup 𝑎 is called the cyclic subgroup of 𝐺 generated by 𝑎.
In case that 𝐺 = 𝑎 , we say that 𝐺 is cyclic and 𝑎 is a generator of
𝐺.
▪ A cyclic group may have many generators.
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Example 6
▪ Consider 𝑈 10 under multiplication modulo 10.
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Example 7
▪ Consider ℤ10 under addition modulo 10.
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Example 8
▪ Consider the group ℤ under addition.
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Reference
Fraleigh, John B. (2002), A First Course in Abstract Algebra, 7th Edn: Addison Wesley
Gallian, Joseph (2004), Contemporary Abstract Algebra, 6th Edn: Brooks Cole
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