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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a [[{{{group property}}}]]. That is, it states that in a warning.png"{{{group property}}}" cannot be used as a page name in this wiki. , every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
[[:Category:Subgroup property implications in {{{group property}}}s|View all subgroup property implications in {{{group property}}}s]] | [[:Category:Subgroup property non-implications in {{{group property}}}s|View all subgroup property non-implications in {{{group property}}}s]] | View all subgroup property implications | View all subgroup property non-implications

[[Category:Subgroup property implications in {{{group property}}}s]]

This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group, is contained in the subgroup
View a complete list of such statements/instances

Contents 1 Statement 1.1 Verbal statement 2 Definitions used 2.1 Supersolvable group 2.2 Self-centralizing subgroup 3 Related facts 4 Facts used 5 Proof 6 References 6.1 Textbook references

Statement

Verbal statement

Suppose G is a supersolvable group and H is maximal among Abelian normal subgroups of G: in other words, H is an Abelian normal subgroup of G, and there is no Abelian normal subgroup of G strictly containing H. Then, H is a self-centralizing subgroup; in other words:

C_G(H) = H

where C_G(H) denotes the centralizer of H.

Definitions used

Supersolvable group

Further information: Supersolvable group

A group G is termed supersolvable if there exists a normal series of G such that all the factor groups are cyclic groups.

Self-centralizing subgroup

Further information: Self-centralizing subgroup

A subgroup H of a group G is termed self-centralizing if . When H is Abelian, this is equivalent to saying C_G(H) = H.

Related facts

• Maximal among Abelian normal implies self-centralizing in nilpotent

Facts used

1. Normality is centralizer-closed: The centralizer of a normal subgroup is normal.
2. Normality satisfies image condition: The image of any normal subgroup under a surjective homomorphism, is a normal subgroup of the image.
3. Supersolvability is quotient-closed: A quotient of a supersolvable group by any normal subgroup is still supersolvable.
4. Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
5. Normality satisfies inverse image condition: The inverse image of a normal subgroup, under any homomorphism, is a normal subgroup.

Proof

Given: A supersolvable group G, a subgroup H that is maximal among Abelian normal subgroups.

To prove: C_G(H) = H

Proof: Suppose C = C_G(H) is the centralizer of H. Since H is normal, so is C.

Since H is Abelian, . Then, since H is normal, we can quotient out by H. Let and . Since , we get .

Now, since G is supersolvable, so is . Thus, the normal subgroup contains a cyclic normal subgroup, say one generated by . Now consider the subgroup generated by H and x in G:

• This is Abelian, since, by definition , so x commutes with all the elements of H, which is itself Abelian.
• This is normal in G, since it is the inverse image of a normal subgroup of under a quotient map.

Hence we have found an Abelian normal subgroup of G properly containing H, a contradiction to the assumption of maximality.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 185, Lemma 3.12 (Section 5.3): proves it in the case that G is a group of prime power order