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Putnam
1997 Putnam
4
Putnam 1997 A4
Putnam 1997 A4
Source:
May 30, 2014
Putnam
function
college contests
Problem Statement
Let
G
G
G
be group with identity
e
e
e
and
ϕ
:
G
→
G
\phi :G\to G
ϕ
:
G
→
G
be a function such that :
ϕ
(
g
1
)
⋅
ϕ
(
g
2
)
⋅
ϕ
(
g
3
)
=
ϕ
(
h
1
)
⋅
ϕ
(
h
2
)
⋅
ϕ
(
h
3
)
\phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3)
ϕ
(
g
1
)
⋅
ϕ
(
g
2
)
⋅
ϕ
(
g
3
)
=
ϕ
(
h
1
)
⋅
ϕ
(
h
2
)
⋅
ϕ
(
h
3
)
Whenever
g
1
⋅
g
2
⋅
g
3
=
e
=
h
1
⋅
h
2
⋅
h
3
g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3
g
1
⋅
g
2
⋅
g
3
=
e
=
h
1
⋅
h
2
⋅
h
3
Show there exists
a
∈
G
a\in G
a
∈
G
such that
ψ
(
x
)
=
a
ϕ
(
x
)
\psi(x)=a\phi(x)
ψ
(
x
)
=
a
ϕ
(
x
)
is a homomorphism. (that is
ψ
(
x
⋅
y
)
=
ψ
(
x
)
⋅
ψ
(
y
)
\psi(x\cdot y)=\psi (x)\cdot \psi(y)
ψ
(
x
⋅
y
)
=
ψ
(
x
)
⋅
ψ
(
y
)
for all
x
,
y
∈
G
x,y\in G
x
,
y
∈
G
)
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