Math Problem Statement

Prove Lagrange's Theorem for a finite group G and its subgroup H.

Solution

To prove Lagrange's Theorem for a finite group GG and its subgroup HH, let's proceed step by step.


Lagrange's Theorem Statement:

If GG is a finite group and HH is a subgroup of GG, then the order (size) of HH divides the order of GG, i.e., HG|H| \,|\, |G|.


Proof Outline:

  1. Cosets and Partitioning of GG:

    • Define left cosets of HH in GG as gH={ghhH}gH = \{ gh \mid h \in H \} for any gGg \in G.
    • The key idea is that the left cosets of HH in GG are disjoint and form a partition of GG. Each element of GG belongs to exactly one coset.
  2. Coset Properties:

    • Property 1: If g1H=g2Hg_1H = g_2H, then g1Hg2H=g1Hg_1H \cap g_2H = g_1H, i.e., two cosets are either identical or disjoint.
    • Property 2: The size of any coset gHgH is equal to H|H|, i.e., gH=H|gH| = |H|, because multiplication by gg is a bijection.
  3. Partition Implies Divisibility:

    • Since cosets partition GG, the total size of GG is the number of distinct cosets (denoted by G:H|G : H|, the index of HH in GG) multiplied by the size of HH.
    • Therefore, G=G:HH|G| = |G : H| \cdot |H|, showing that H|H| divides G|G|.

Detailed Proof:

  1. Establishing Coset Properties:

    • If tsHt \in sH, then sH=tHsH = tH (this ensures cosets are well-defined).
    • If tsHt \notin sH, then sHtH=sH \cap tH = \emptyset (cosets are either disjoint or identical).
    • The size of each coset sHsH is H|H| because the map hshh \mapsto sh for hHh \in H is bijective.
  2. Partitioning:

    • The disjoint union of all cosets of HH gives GG: G=gRgHG = \bigcup_{g \in R} gH, where RR is a set of representatives for distinct cosets.
    • The total number of cosets is G:H|G : H|, the index of HH in GG.
  3. Concluding Divisibility:

    • From the partitioning, G=G:HH|G| = |G : H| \cdot |H|, proving that HG|H| \,|\, |G|.

Final Answer for Question 4:

By the above steps, we have shown that for a finite group GG and a subgroup HH, the order of HH divides the order of GG.


Would you like a more detailed explanation of any step? Or additional examples?

Here are 5 follow-up questions you might find helpful:

  1. What is the significance of cosets in group theory beyond Lagrange's theorem?
  2. Can Lagrange's theorem apply to infinite groups?
  3. How do right cosets differ from left cosets, and does it affect the theorem?
  4. Can Lagrange's theorem provide information about possible subgroup sizes?
  5. What are practical examples of applying Lagrange's theorem in algebra?

Tip: Always verify that the subgroup HH is closed under group operations and that it contains the identity element to ensure it satisfies the subgroup criteria.

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Math Problem Analysis

Mathematical Concepts

Group Theory
Cosets
Finite Groups

Formulas

|G| = |G : H| * |H|

Theorems

Lagrange's Theorem

Suitable Grade Level

Undergraduate Mathematics