a poem, a strategic lemma


Lemma

\cdot  p\in{\Bbb{P}}

\cdot |G|=p^mr , p\nmid r

\cdot \forall H<G

\cdot \forall Q\in {\rm p\!-\!SS}_G

\Longrightarrow

\bullet \exists g\in G such that H\cap gQg^{-1}\in {\rm p\!-\!SS}_H.

Proof:

By employing the Double Coset Counting Formula we have |G|=\sum_a\frac{|H|\ |Q|}{|H\cap aQa^{-1}|}, and since p\nmid [G:Q] then \exists b\in\{a\} such that p\nmid [H:H\cap bQb^{-1}].

But H\cap bQb^{-1}<bQb^{-1} so |H\cap bQb^{-1}|=p^l\ ,\ \exists l\in{\Bbb{N}}, hence, having p\nmid\frac{|H|}{p^l}, this implies that |H|=p^l\alpha, where p\nmid\alpha\ ,\ \exists \alpha\in{\Bbb{N}}.

Then, for H\cap bQb^{-1}<H with |H\cap bQb^{-1}|=p^l we deduce it is {\rm p\!-\!SS}_H

\Box

DCCF01

double coset counting formula

 

3 Comments

Filed under algebra, group theory, math, word algebra

3 responses to “a poem, a strategic lemma

  1. Reblogged this on ianmarqz and commented:
    reblog test

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