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Contest of the Centers of Excellency of Suceava
2012 Centers of Excellency of Suceava
1
Group theory operations
Group theory operations
Source:
October 31, 2019
group theory
abstract algebra
Problem Statement
Let be a natural number
n
≥
2
,
n\ge 2,
n
≥
2
,
a group
G
G
G
and two elements of it
e
1
,
e
2
e_1,e_2
e
1
,
e
2
such that
e
2
e
1
x
=
x
e
2
e
1
,
e_2e_1x=xe_2e_1,
e
2
e
1
x
=
x
e
2
e
1
,
for any element
x
x
x
of
G
.
G.
G
.
Prove that
(
e
1
x
e
2
)
n
=
e
1
x
n
e
2
,
\left( e_1xe_2 \right)^n =e_1x^ne_2,
(
e
1
x
e
2
)
n
=
e
1
x
n
e
2
,
for any element
x
x
x
of
G
,
G,
G
,
if and only if
e
2
e
1
=
(
e
2
e
1
)
n
.
e_2e_1=\left( e_2e_1\right)^n.
e
2
e
1
=
(
e
2
e
1
)
n
.
Ion Bursuc
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