Give a subgroup with an identity such that this identity does not hold in the bigger group.
$begingroup$
In universal algebra, I am trying to show a counterexample using groups for the converse of "If an identity $s=t$ holds in $mathcal{L}$-algebra $mathcal{A}$, and $mathcal{B}$ is a subalgebra of $mathcal{A}$, then this identity also holds in $mathcal{B}$".
My counterexample is the group $mathbb{Z}_{n}$ with subgroup $U(n)=left {xin mathbb{Z}_{n}:gcd(x,n)=1 right }$. The identity $gcd(x,n)=1$ only holds in $U(n)$. Is this a valid counterexample?
abstract-algebra group-theory universal-algebra
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
$endgroup$
|
show 10 more comments
$begingroup$
In universal algebra, I am trying to show a counterexample using groups for the converse of "If an identity $s=t$ holds in $mathcal{L}$-algebra $mathcal{A}$, and $mathcal{B}$ is a subalgebra of $mathcal{A}$, then this identity also holds in $mathcal{B}$".
My counterexample is the group $mathbb{Z}_{n}$ with subgroup $U(n)=left {xin mathbb{Z}_{n}:gcd(x,n)=1 right }$. The identity $gcd(x,n)=1$ only holds in $U(n)$. Is this a valid counterexample?
abstract-algebra group-theory universal-algebra
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
$endgroup$
2
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
1
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
4
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
1
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53
|
show 10 more comments
$begingroup$
In universal algebra, I am trying to show a counterexample using groups for the converse of "If an identity $s=t$ holds in $mathcal{L}$-algebra $mathcal{A}$, and $mathcal{B}$ is a subalgebra of $mathcal{A}$, then this identity also holds in $mathcal{B}$".
My counterexample is the group $mathbb{Z}_{n}$ with subgroup $U(n)=left {xin mathbb{Z}_{n}:gcd(x,n)=1 right }$. The identity $gcd(x,n)=1$ only holds in $U(n)$. Is this a valid counterexample?
abstract-algebra group-theory universal-algebra
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
$endgroup$
In universal algebra, I am trying to show a counterexample using groups for the converse of "If an identity $s=t$ holds in $mathcal{L}$-algebra $mathcal{A}$, and $mathcal{B}$ is a subalgebra of $mathcal{A}$, then this identity also holds in $mathcal{B}$".
My counterexample is the group $mathbb{Z}_{n}$ with subgroup $U(n)=left {xin mathbb{Z}_{n}:gcd(x,n)=1 right }$. The identity $gcd(x,n)=1$ only holds in $U(n)$. Is this a valid counterexample?
abstract-algebra group-theory universal-algebra
abstract-algebra group-theory universal-algebra
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
asked Jan 15 at 20:05
numericalorangenumericalorange
1,850312
1,850312
2
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
1
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
4
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
1
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53
|
show 10 more comments
2
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
1
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
4
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
1
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53
2
2
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
1
1
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
4
4
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
1
1
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53
|
show 10 more comments
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2
$begingroup$
Can $gcd(x,n)=1$ be expressed in the language of Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:07
1
$begingroup$
Since when has this $U(n)$ been a subgroup of $Bbb Z_n$? It's not closed under addition.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 20:08
$begingroup$
@LordSharktheUnknown My mistake! I will try to find another example that actually makes sense.
$endgroup$
– numericalorange
Jan 15 at 20:09
4
$begingroup$
Your idea will actually work with the trivial subgroup of any group, since all identities hold in the trivial group and not in any other group.
$endgroup$
– Tobias Kildetoft
Jan 15 at 20:32
1
$begingroup$
There is ton of examples really. For instance, you can also take ${0,2}leq mathbb{Z}_4$. The subgroup satisfies $x+xapprox 0$, but this does not hold in $mathbb{Z}_4$, since $1+1neq 0$.
$endgroup$
– OnDragi
Feb 25 at 16:53