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
• Frobenius automizer criterion for p nilpotence Corollary
• Frobenius semidirect product decomposition 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
• A solvable group that is not an M-group Counterexample
• A transitive action need not be Frobenius Counterexample
• Cyclic sylow does not alone imply a normal p complement 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
• Normal p complement and p nilpotent group Definition
• P residual of a finite group Definition
• The p-core Oₚ(G) as the largest normal p-subgroup Definition
• The p-prime core of a finite group Definition
• [G:G]=1 and, for finite G, [G:{e}]=|G| Example
• A₄ has no subgroup of order 6 Example
• A5 as the smallest nonabelian simple group Example
• Affine linear Frobenius groups over finite fields Example
• Class group of Q(sqrt 10) Example
• Frobenius normal two complement for S₃ Example
• S₃ as a Frobenius group Example
• Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
• The B2 Davis complex is an octagon whose boundary is the Coxeter complex circle Example
• The Coxeter complex of A₃: a triangulation of the sphere and the residue of a proper parabolic Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers 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 right-angled cube Davis complex and its boundary 2-sphere Example

…and 72 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)