Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$

Depends on

• In a finite group, the subgroup, every coset and the set of cosets are finite
• The left cosets of a subgroup partition the group
• Every left or right coset of $H$ is equinumerous with $H$
• The coset set $G/H$ and the index $[G:H]$ of a subgroup
• The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity
• The cardinality $\lvert A\rvert$ of a finite set
• The sum rule: a finite disjoint union is finite with $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$, and a sum over a finite index set splits along a partition
• The sum $\sum_{i \in S} a_i$ over a finite index set, and its product form
• Divisibility in $\mathbb{Z}$: $d \mid a$ when $a = dq$ for some integer $q$
• The naturals embed in the integers

Used by

• A noncentral element of an extraspecial p-group has centraliser of index p Corollary
• An extraspecial p-group has order p¹⁺²ⁿ for some n≥1 Corollary
• An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
• Every subgroup of index p in a finite p-group is normal Corollary
• For a finite group, uniquely divisible coefficients have trivial first cohomology Corollary
• For an odd prime p, ℚ(ζₚ) has exactly one intermediate field of degree two over ℚ Corollary
• For K≤ H≤ G with G finite, [G:K]=[G:H][H:K] Corollary
• For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
• If [G:N] is finite then |G/N|=[G:N]; for finite G this equals |G|/|N| Corollary
• Orbit-stabiliser cardinality: |G· x|=[G:Gₓ] whenever either side is finite, and |G|=|Gₓ| |G· x| for finite G Corollary
• The class equation of Sₙ is n!=∑_∑ k cₖ=n n!/∏ₖ k^cₖcₖ! Corollary
• The order of a finite group is the product of the orders of its composition factors Corollary
• The order of every element of a finite group divides the order of the group Corollary
• There are exactly two isomorphism classes of groups of order 105 Corollary
• 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
• The product set HK of two subgroups need not be a subgroup Counterexample
• The subgroup ⟨(1 2 3),(1 2)(3 4)⟩≤ S₄ has order 12 but no subgroup of order 6, so Cauchy's theorem does not extend to composite divisors Counterexample
• The p-core Oₚ(G) as the largest normal p-subgroup Definition
• [G:G]=1 and, for finite G, [G:{e}]=|G| Example
• A₄ has no subgroup of order 6 Example
• Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
• The Heisenberg group of order 27 has exponent 3 and thirteen subgroups of order 3 Example
• The modular group of order 27 has exponent 9 and exactly three cyclic subgroups of order 9 Example
• The subgroup orders in Sym({1,2,3}) are 1,2,3 and 6 Example
• The three maximal abelian subgroups of Dih(C₄) have order four, as the general bound predicts Example
• False statement: every divisor of the order of a finite group occurs as a subgroup order False statement
• FALSE: every finite abelian group is Gal(ℚ(μₙ)/ℚ) for some n False statement
• A finite cyclic group has exactly one subgroup of each order dividing its own Lemma
• A finite product of normal p-subgroups is a normal p-subgroup Lemma
• A normal Hall subgroup presents the ambient group as an extension of coprime orders Lemma
• A product formula for the number of square roots of the identity in a central product of extraspecial 2-groups Lemma
• A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient Lemma
• Distinct normal Sylow subgroups centralize one another Lemma
• Every subgroup of a finite p-group has order a power of p Lemma
• For primes p<q, nontrivial actions of Cₚ on C_q exist exactly when p∣(q-1) and are unique up to automorphisms Lemma
• If p<q are primes and |G|=pq, then G has a normal subgroup of order q Lemma
• Q₈∘ Q₈ congDih(C₄)circDih(C₄) Lemma
• The transitive subgroups of S₄ and their action on the three pairings Lemma
• Two elements of an extraspecial p-group with nontrivial commutator generate an extraspecial subgroup of order p³ Lemma
• Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition

…and 35 more results.

Dependency tree · two levels

49 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• UCL lecture notes, Cosets and Lagrange's theorem (standard reference, not scraped)
• Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.2: Lagrange's Theorem (standard reference, not scraped)