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|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7

Source:

August 28, 2010

abstract algebrainequalitiesgroup theorysuperior algebrasuperior algebra unsolved

Problem Statement

Let

A

be an abelian additive group such that all nonzero elements have infinite order and for each prime number

p

we have the inequality

|A/pA|\leq p

, where

pA = \{pa |a \in A\}

pa = a+a+\cdots+a

(where the sum has

p

summands) and

|A/pA|

is the order of the quotient group

A/pA

(the index of the subgroup

pA

).
Prove that each subgroup of

A

of finite index is isomorphic to

A