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groups and proper subgroups

Source: Romanian Nationals RMO 2005 - grade 12, problem 2

March 31, 2005

group theoryabstract algebrafunctionLaTeXsuperior algebrasuperior algebra unsolved

Problem Statement

Let

G

be a group with

m

elements and let

H

be a proper subgroup of

G

with

n

elements. For each

x\in G

we denote

H^x = \{ xhx^{-1} \mid h \in H \}

and we suppose that

H^x \cap H = \{e\}

, for all

x\in G - H

(where by

e

we denoted the neutral element of the group

G

). a) Prove that

H^x=H^y

if and only if

x^{-1}y \in H

; b) Find the number of elements of the set

\bigcup_{x\in G} H^x

as a function of

m

and

n

. Calin Popescu

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