(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

suppose that H and K are subgroups of a group G such that K is a proper subgroup of H which is a proper subgroup of G and suppose (H : K) and (G : H) are both finite. Then (G : K) is finite, and (G : K) = (G : H)(H : K).

**that is to say that the proof must hold for infinite groups as well**

notation- (G : K) = |G|/|K| is the index of K in G

2. Relevant equations

Lagrange's Theorem - b/c G is finite implies that there is a finite subgroup in G (i.e. H) whose order divides that of G's.

**there is no mention that the group G in question is considered to be an abelian group.**

3. The attempt at a solution

if we say that {(a_i)H | i = 1, ... , r } is the collection of distinct left cosets of H in G and {(b_j)K | j = 1, ... , s } is the collection of distinct left cosets of K in H.

then in order to conclude the proof I have to show that:

{(a_i)(b_j)K | i = 1, ... , r; j = 1,...,s } is the collection of distinct left cosets of K in G.

i was not sure about a method of approach that came to me. so i was thinking of a few ways to solve it, but im not sure of the right one if any of them are correct.

*1-that is (a_i)H is the number of distinct left cosets of H in G. so b/c |G| is finite then |G| must either be prime or not prime. if |G| is prime then let |G| = p and let |H| = m where m is an element ofNso by Lagrange's Theorem we know that m divides p. and b/c p is prime then we know that m = p. but this is not true because H is a proper subset of G so p > m... then |G| must not be prime let |G| = y in that case then we can find an element "x" in theNwhere |H| = x such that x divides y and x < y.

im not sure if this is at all the correct way to approach it because i am having trouble relating this to the distinct cosets of H in G

i would really appreciate some help on the matter.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Lagrange, cosets, indicies

**Physics Forums | Science Articles, Homework Help, Discussion**