Alphabeta Math

Cosets, Index and Lagrange's Theorem

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✓ 13 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Left and right cosets gH and Hg of a subgroup

Lemma

x∈ aH iff a⁻¹x∈ H, and aH=bH iff a⁻¹b∈ H

Lemma

The left cosets of a subgroup partition the group

Lemma

Every left or right coset of H is equinumerous with H

Lemma

Inversion induces a bijection gH↦ Hg⁻¹ from left cosets to right cosets

Definition

The coset set G/H and the index [G:H] of a subgroup

Lemma

In a finite group, the subgroup, every coset and the set of cosets are finite

Theorem

Lagrange's theorem: |G|=[G:H]|H| for every subgroup H of a finite group G

Corollary

The order of every element of a finite group divides the order of the group

Corollary

g^|G|=e for every element g of a finite group G

Corollary

A finite group of prime order is cyclic and every nonidentity element generates it

Corollary

For K≤ H≤ G with G finite, [G:K]=[G:H][H:K]

Corollary

[G:H]=1 if and only if H=G

Theorem

Euler's theorem: if n≥1 and gcd(a,n)=1, then a^φ(n)≡1 (mod n)

Theorem

Fermat's little theorem: for prime p, p∤ a implies aᵖ⁻¹≡1 (mod p), and always aᵖ≡ a (mod p)

Sources