Poincare's theorem
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This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This article gives the statement, and possibly proof, of a subgroup property (i.e., subgroup of finite index) satisfying a subgroup metaproperty (i.e., normal core-closed subgroup property)
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Statement
Verbal statement
If a group (possibly infinite) has a subgroup of finite index, say , then that subgroup contains a normal subgroup of finite index, where the index is at most . In fact, we can choose the normal subgroup such that its index is a multiple of and a divisor of .
(The subgroup that we choose here is the normal core of the original subgroup).
Symbolic statement
Suppose a group , a subgroup of index . Then, contains a subgroup that is normal in , with index in at most . In fact, we can choose such that:
.
(The subgroup that we choose here is the normal core of the original subgroup).
Related facts
Other facts about normal subgroups and index using the idea of the action on the coset space
- Conjugate-intersection index theorem: This gives a bound on the intersection of finitely many conjugate subgroups
- Subgroup of index two is normal
- Subgroup of least prime index is normal
Analogous facts for characteristic subgroups
Facts used
- Group acts on left coset space of subgroup by left multiplication
- First isomorphism theorem
- Lagrange's theorem
- Index is multiplicative
Proof
In group action language
Given: A group , a subgroup of index .
To prove: contains a subgroup that is normal in , with index at most . Further, we can choose such that .
Proof:
- Consider the action of by left multiplication on the left coset space (fact (1)). This gives a homomorphism from .
- Let be the kernel of . Then is normal and : The kernel is precisely the intersection of the isotropies of all the points of ; equivalently, it is the intersection of all conjugates of . In particular, . is also the normal core of .
- The index of is at most . In fact, it divides : By the first isomorphism theorem (fact (2)), is isomorphic to the image , which is a subgroup of . Thus, the index of is at most . In fact, fact (3) (Lagrange's theorem) yields that the order of divides . Hence, the index also divides .
- The index of is a multiple of : This follows from fact (4), applied to the groups .
References
Textbook references
- Topics in Algebra by I. N. Herstein, ^{More info}, Page 48, Exercise 20