Below is a clear, simple, and exam-oriented explanation of Subgroups and Cosets — perfect for BCA/MCA/Engineering/Discrete Mathematics or Abstract Algebra exams.
⭐ SUBGROUPS AND COSETS
Subgroups and cosets are essential concepts in Group Theory, used to understand the structure of groups and prove important theorems like Lagrange’s Theorem.
⭐ 1. SUBGROUPS
A subgroup is a smaller group inside a bigger group.
⭐ Definition
A subset ( H ) of a group ( G ) is a subgroup if:
1️⃣ (H) is non-empty
2️⃣ (H) is closed under the group operation
3️⃣ (H) contains the identity element of G
4️⃣ Every element of (H) has an inverse in (H)
If these hold, then (H) is a subgroup of (G).
Notation:
[
H \le G
]
⭐ Subgroup Test (Most important shortcut)
A non-empty subset ( H \subseteq G ) is a subgroup if and only if:
[
a, b \in H \Rightarrow ab^{-1} \in H
]
This automatically gives closure, identity, and inverses.
⭐ Examples of Subgroups
✔ Example 1: Even integers
Group: ( (\mathbb{Z}, +) )
Subgroup: ( 2\mathbb{Z} = {\dots,-4,-2,0,2,4,\ldots} )
- Closed under +
- Contains 0
- Inverses exist
→ Subgroup
✔ Example 2: Multiples of n in integers
Subgroup of (ℤ, +):
[
n\mathbb{Z} = {0, n, 2n, 3n, \ldots}
]
✔ Example 3: Real numbers under multiplication
Group: ( (\mathbb{R}^\times, \times) )
Subgroup: positive real numbers ( \mathbb{R}^+ )
✔ Example 4: Symmetric Group ( S_3 )
Subgroup:
[
H = {e, (1\ 2)}
]
⭐ Non-Examples
✘ Natural numbers under addition
Not all inverses present → NOT subgroup.
✘ Set {1, 2, 3} under addition mod 7
Closure fails → NOT subgroup.
⭐ 2. COSETS
Cosets are used to partition a group into equal-sized pieces.
⭐ Definition
Let ( G ) be a group and ( H \le G ) a subgroup.
For any ( a \in G ):
Left coset of H with representative a:
[
aH = {ah : h \in H}
]
Right coset of H with representative a:
[
Ha = {ha : h \in H}
]
In abelian groups,
[
aH = Ha
]
⭐ Examples
✔ Example 1: (ℤ, +) with subgroup 3ℤ
Group: ℤ
Subgroup: ( 3\mathbb{Z} = {\ldots, -6, -3, 0, 3, 6, \ldots} )
Left coset (and right coset):
[
1 + 3\mathbb{Z} = {\ldots,-5,-2,1,4,7,10,\ldots}
]
[
2 + 3\mathbb{Z} = {\ldots,-4,-1,2,5,8,11,\ldots}
]
There are exactly 3 cosets:
0+3ℤ, 1+3ℤ, 2+3ℤ.
✔ Example 2: Symmetric Group S₃
Let
[
H = {e, (1 2)}
]
Left coset of (1 3):
[
(1 3)H = {(1 3), (1\ 3)(1\ 2)}
= {(1 3), (1\ 2\ 3)}
]
⭐ Key Properties of Cosets
1️⃣ Cosets partition the group
Every element of G belongs to exactly one left coset of H.
2️⃣ All cosets have the same number of elements as H.
This is used in Lagrange’s Theorem.
3️⃣ Two cosets are either:
- identical, or
- disjoint
4️⃣ aH = bH iff ( a^{-1}b \in H )
⭐ Index of a Subgroup
Number of distinct left cosets of H in G:
[
[G : H]
]
If G is finite:
[
|G| = [G:H] \cdot |H|
]
This is Lagrange’s Theorem.
⭐ 3. LAGRANGE’S THEOREM (related to cosets)
If ( G ) is a finite group and ( H \le G ), then:
[
|H| \mid |G|
]
Because cosets partition G.
Important consequences:
- Order of an element divides order of group.
- Any group of prime order is cyclic.
- Subgroups have sizes dividing |G|.
⭐ Visualization of Cosets
Think of a group as a cake.
A subgroup cuts the cake into equal slices called cosets.
Each slice (coset):
- same size
- disjoint
- covers the entire cake (group)
⭐ Quick Exam-Oriented Summary
Subgroup Definition:
A subset closed under operation and inverses.
Subgroup Test:
[
a, b \in H \Rightarrow ab^{-1} \in H
]
Coset Definition:
Left coset: (aH = {ah:h\in H})
Right coset: (Ha = {ha:h\in H})
Properties:
- Same size as H
- Partition the group
- Either identical or disjoint
Lagrange’s Theorem:
[
|G| = |H| \cdot [G:H]
]