Second isomorphism theorem
This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article is about an isomorphism theorem in group theory.
View a complete list of isomorphism theorems| Read a survey article about the isomorphism theorems
Name
This result is termed the second isomorphism theorem or the diamond isomorphism theorem (the latter name arises because of the diamond-like shape that can be used to describe the theorem).
Statement
General statement
Suppose is a group, and
are two subgroups of
such that
normalizes
; in other words
for every
. Then,
is a group, with
(i.e.,
is a normal subgroup of
), and
(i.e.
is a normal subgroup of
), and:
Particular case
Suppose is a group,
is a normal subgroup, and
an arbitrary subgroup, such that
. Then:
Definitions used
Term | Relevant definitions(s) |
---|---|
normal subgroup | A subgroup ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
isomorphism of groups | Given groups ![]() ![]() ![]() ![]() ![]() ![]() |
quotient group | For a normal subgroup ![]() ![]() ![]() ![]() |
Particular cases
Specific possibilities for the relationship between
and 
Note that for each of these, if we have , then the statement made for
applies to
.
Situation | Interpretation |
---|---|
![]() |
In this case, ![]() ![]() |
![]() |
In this case, ![]() ![]() |
![]() |
In this case, ![]() ![]() ![]() ![]() |
![]() ![]() ![]() |
In this case, ![]() ![]() ![]() ![]() ![]() |
Specific possibilities for
and the location of
and
within
Nature of ![]() |
Nature/parameters for ![]() ![]() |
Interpretation of ![]() ![]() |
Interpretation of ![]() ![]() |
Fact in this context that corresponds to second isomorphism theorem |
---|---|---|---|---|
group of integers ![]() |
![]() ![]() |
If ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
The arithmetic fact that ![]() |
vector space ![]() ![]() |
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
The dimension of ![]() ![]() ![]() ![]() |
The isomorphism between these (which turns out to also be a vector space isomorphism) preserves dimension, and we get ![]() ![]() |
![]() ![]() |
There are subsets ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
The fact that ![]() ![]() |
Related facts
Facts about normal subgroups
- Normality satisfies transfer condition: If
are subgroups and
is normal in
, then
is normal in
.
- Normality satisfies intermediate subgroup condition: If
are groups and
is normal in
, then
is normal in
.
General version of the result
- Product formula: The set-theoretic version of the product formula establishes a bijection which is the same as the bijection of the second isomorphism theorem, but without the conditions of normality. The bijection is purely at the set-theoretic level.
Facts used
Proof
Given: A group , subgroups
such that
for all
.
To prove: is a group,
is normal in
,
is normal in
, and
.
Proof:
is a subgroup of 
Condition | In symbols | Justification |
---|---|---|
Closure under multiplication | For ![]() ![]() |
Observe that ![]() ![]() ![]() |
Identity element | The identity element of ![]() ![]() |
The identity element is in both ![]() ![]() ![]() |
Inverses | Given ![]() ![]() |
![]() ![]() ![]() |
is normal in 
Observe that, by the condition, for any
. Thus, for
and
, we have
. Thus,
is normal in
.
is normal in 
We now prove that is normal in
. Pick
. Then
such
is a subgroup. Also, since
for any
, we have
, so
. Thus,
, and we get that
is normal in
.
Definition of isomorphism and proof that it works
Finally, define the isomorphism between
and
as follows:
.
We check that this satisfies all the required conditions:
Condition | Verification |
---|---|
Sends cosets to cosets | Note first that if two elements are in the same coset of ![]() ![]() ![]() ![]() |
Well-defined with specified domain and co-domain | Further, if ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Injective | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Surjective | Any left coset of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Homomorphism | Suppose ![]() ![]() |
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 97, Theorem 18 of Section 3.3, More info
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, Page 236, Exercise 7 of Miscellaneous Problems, More info