Can a group be isomorphic to one of its quotients?

In summary, it is not always the case that there is a subgroup isomorphic to every quotient of a group G/N. This is true even for infinite groups, as shown by the example of ##\mathbb{Z}\times\mathbb{Z}##. Additionally, even simple groups may have non-trivial subgroups but no non-trivial quotient. An example of this is the permutation group on 5 elements, ##S_5##, which has the alternating subgroup that cannot be a quotient.
  • #1
WWGD
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Of course it must be an infinite group, otherwise |G/N|=|G|/|N| and then {e} is the only ( and trivial) solution. I understand there is a result that for every quotient Q:=G/N there is a subgroup H that is isomorphic to Q. Is that the case?
 
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  • #2
How about an infinite product ##\mathbb{Z}\times\mathbb{Z}\times...## and quotient out by the first factor?

WWGD said:
I understand there is a result that for every quotient Q:=G/N there is a subgroup H that is isomorphic to Q. Is that the case?
No, there is no subgroup of ##\mathbb{Z}## that is isomorphic to ##\mathbb{Z}/2\mathbb{Z}.##
 
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  • #3
Infrared said:
How about an infinite product ##\mathbb{Z}\times\mathbb{Z}\times...## and quotient out by the first factor?No, there is no subgroup of ##\mathbb{Z}## that is isomorphic to ##\mathbb{Z}/2\mathbb{Z}.##
Thanks. Sorry, I believe the result I quoted (may)apply to infinite groups.
 
  • #4
WWGD said:
Thanks. Sorry, I believe the result I quoted (may)apply to infinite groups.
No, even if the quotient is infinite, it is still false. There is no subgroup of ##\mathbb{Z}\times\mathbb{Z}## that is isomorphic to ##\left(\mathbb{Z}\times\mathbb{Z}\right)/\left(\{0\}\times 2\mathbb{Z}\right)\cong\mathbb{Z}\times\left(\mathbb{Z}/2\mathbb{Z}\right).##
 
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  • #5
Re the lack of correspondence between subgroups and quotient groups, I though of another argument: just take a simple group. It will have non-trivial subgroups but no non-trivial quotient. Maybe simplest vase is ##S_5##, the permutation group on 5 elements . It has the alternating subgroup, which cannot be a quotient by cardinality reasons.
 

Related to Can a group be isomorphic to one of its quotients?

1. Can a group be isomorphic to one of its quotients?

Yes, it is possible for a group to be isomorphic to one of its quotients. This means that there exists a one-to-one correspondence between the elements of the group and the elements of its quotient, such that the group operation is preserved.

2. What is an isomorphism?

An isomorphism is a bijective homomorphism between two groups, meaning it is a function that preserves the group structure and is both one-to-one and onto.

3. How can we prove that a group is isomorphic to one of its quotients?

To prove that a group is isomorphic to one of its quotients, we can show that there exists a bijective homomorphism between the two groups. This can be done by constructing a mapping between the elements of the group and the elements of its quotient, and then showing that this mapping preserves the group operation.

4. Is every group isomorphic to one of its quotients?

No, not every group is isomorphic to one of its quotients. For a group to be isomorphic to one of its quotients, the quotient group must have the same structure as the original group. This means that the quotient must have the same number of elements and the same group operation as the original group.

5. What are some real-world applications of isomorphism in group theory?

Isomorphism in group theory has many real-world applications, such as in cryptography, coding theory, and network analysis. For example, isomorphic groups can be used to encrypt and decrypt messages, encode and decode data, and identify patterns in complex networks.

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