Mixing
How to determine if a surjective homomorphism exists between two groups?
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I've been racking my brain for some property of homomorphisms to do with cardinality, or something like that, but I basically have no idea what I'm doing. Any pointers?
group-theory permutations group-homomorphism
asked 4 hours ago
cal
361
add a comment |Â
up vote
3
down vote
favorite
My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I've been racking my brain for some property of homomorphisms to do with cardinality, or something like that, but I basically have no idea what I'm doing. Any pointers?
group-theory permutations group-homomorphism
asked 4 hours ago
cal
361
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I've been racking my brain for some property of homomorphisms to do with cardinality, or something like that, but I basically have no idea what I'm doing. Any pointers?
group-theory permutations group-homomorphism
asked 4 hours ago
cal
361
My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I've been racking my brain for some property of homomorphisms to do with cardinality, or something like that, but I basically have no idea what I'm doing. Any pointers?
group-theory permutations group-homomorphism
group-theory permutations group-homomorphism
asked 4 hours ago
cal
361
asked 4 hours ago
cal
361
asked 4 hours ago
cal
361
asked 4 hours ago
cal
361
asked 4 hours ago
cal
361
361
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.
The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.
answered 4 hours ago
hunter
13.7k22337
add a comment |Â
up vote
1
down vote
A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.
answered 3 hours ago
Oiler
1,9041820
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.
The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.
answered 4 hours ago
hunter
13.7k22337
add a comment |Â
up vote
4
down vote
Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.
The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.
answered 4 hours ago
hunter
13.7k22337
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.
The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.
answered 4 hours ago
hunter
13.7k22337
Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.
The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.
answered 4 hours ago
hunter
13.7k22337
answered 4 hours ago
hunter
13.7k22337
answered 4 hours ago
hunter
13.7k22337
answered 4 hours ago
hunter
13.7k22337
13.7k22337
add a comment |Â
add a comment |Â
up vote
1
down vote
A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.
answered 3 hours ago
Oiler
1,9041820
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
add a comment |Â
up vote
1
down vote
A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.
answered 3 hours ago
Oiler
1,9041820
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.
answered 3 hours ago
Oiler
1,9041820
A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.
answered 3 hours ago
Oiler
1,9041820
answered 3 hours ago
Oiler
1,9041820
answered 3 hours ago
Oiler
1,9041820
answered 3 hours ago
Oiler
1,9041820
1,9041820
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
add a comment |Â
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
And use this in tandem with Lagrange: divisibility issues will often show it is impossible to have certain surjections
– Randall
3 hours ago
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2961456%2fhow-to-determine-if-a-surjective-homomorphism-exists-between-two-groups%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
