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Vojtěch Jarník IMC
2011 VJIMC
Problem 4
syseq in group, 3 eqns/variables
syseq in group, 3 eqns/variables
Source: VJIMC 2011 1.4
June 1, 2021
group theory
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be elements of finite order in some group. Prove that if
a
−
1
b
a
=
b
2
a^{-1}ba=b^2
a
−
1
ba
=
b
2
,
b
−
2
c
b
2
=
c
2
b^{-2}cb^2=c^2
b
−
2
c
b
2
=
c
2
, and
c
−
3
a
c
3
=
a
2
c^{-3}ac^3=a^2
c
−
3
a
c
3
=
a
2
then
a
=
b
=
c
=
e
a=b=c=e
a
=
b
=
c
=
e
, where
e
e
e
is the unit element.
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