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Groups, Subgroups & Cosets

Useful preparation: Logic, Sets & Relations

For a long equation, swipe sideways inside its formula box to see every term.

Meaning & notation

Groups describe operations that combine and undo. Subgroups collect self-contained operations; cosets divide the group into equally sized translated copies.

Group and abelian group: A group is a nonempty set with a closed associative binary operation, an identity e, and a two-sided inverse for every element. It is abelian when the operation also commutes.
Subgroup and generated subgroup: A subgroup H is a subset that is itself a group under the inherited operation. The subgroup generated by a, written ⟨a⟩, contains all integer powers of a, or integer multiples in additive notation.
Coset and index: The left coset gH consists of gh for h in H. The index $\displaystyle \left[G : H\right]$ counts distinct left cosets. In additive groups, write $\displaystyle g + H$ instead.

Conditions before calculation

A subgroup candidate must be nonempty. It suffices to show that $\displaystyle ab^{-1}$ belongs to it whenever a and b do; in additive notation this becomes closure under differences.
Lagrange’s numerical statement $\displaystyle \left|G\right| = \left[G : H\right]$ $\displaystyle \left|H\right|$ concerns finite G. Divisibility of group sizes is a necessary condition for a subgroup, not a general existence theorem.
Cosets form a quotient group under their natural multiplication only if H is normal: gHg⁻¹=H for every g. Every subgroup of an abelian group satisfies this condition.

Why the method works

1. Separate the operation from the elements

The same underlying set may form a group under one operation and fail under another. In modular addition the identity is the zero class and the inverse of a is −a.

2. Derive the order formula

In $\mathbb{Z}/n\mathbb{Z}$ under addition, the order of a is the smallest positive k for which n divides ka. Dividing n and a by their greatest common divisor reveals that smallest k.

\operatorname{ord}([a])=\frac n{\gcd(n,a)}

3. Count equal blocks

Multiplication by g maps H bijectively onto gH; multiplication by $\displaystyle g^{-1}$ reverses it. Cosets are either disjoint or equal, so counting their elements gives Lagrange’s formula.

|G|=[G:H]\cdot|H|

Worked examples

EXAMPLE 1

In the additive group $\mathbb{Z}/18\mathbb{Z}$ , find the subgroup generated by 6 and its index.

Repeated addition produces the residue classes 0,6,12, then returns to 0.
These three classes form the generated subgroup; the order formula agrees because $\displaystyle \operatorname{gcd}\, \left(18, 6\right) = 6$ .
Each coset contains three elements. Dividing 18 by 3 gives six cosets, represented by 0 through 5.

Conclusion: The subgroup is {0,6,12}, the element order is 3, and the index is 6.

EXAMPLE 2

In $\mathbb{Z}/20\mathbb{Z}$ , let $\displaystyle H = \left\{0, 5, 10, 15\right\}$ . Decide whether $\displaystyle 3 + H$ and $\displaystyle 8 + H$ are different cosets.

Subtracting any two listed multiples of 5 gives another such class modulo 20, so H is a subgroup.
The difference $\displaystyle 8 - 3 = 5$ lies in H, which means translation by 8 reaches the same block as translation by 3.
Listing confirms the conclusion.
$3+H=\{3,8,13,18\}=8+H$

Conclusion: The two expressions name the same coset, not two different blocks.

Pitfalls & counterexamples

Tempting claim: Every coset is itself a subgroup.

Correction: Only the coset containing the identity equals H and is a subgroup.

Test the claim: In $\mathbb{Z}/20\mathbb{Z}$ , the coset $\displaystyle 3 + H$ above does not contain 0, so it cannot be an additive subgroup.

Check your understanding

What is the additive order of 8 in $\mathbb{Z}/20\mathbb{Z}$ ?

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Compare your reasoning

The smallest positive k with 20 dividing 8k is 5; equivalently, $\displaystyle \frac{20}{\operatorname{gcd}}\, \left(20, 8\right) = 5$ .

Connect this idea

Cosets are equivalence classes. Their equal sizes prepare kernel-image counting; normality makes quotient multiplication well-defined.

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Reference reading

Use these texts to study the underlying definitions and theorems in greater depth. The explanations and examples above are written for Numerive.