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This article describes a result about finite groups where the isomorphism types of the Sylow subgroups implies certain properties of the whole group. Note that all p-Sylow subgroups for a given prime p are conjugate and hence are isomorphic, so the statement makes sense.
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This article gives a proof/explanation of the equivalence of multiple definitions for the term Z-group
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Contents 1 Statement 2 Related facts 2.1 Applications 3 Facts used 4 Proof using advanced methods 4.1 Showing that the group is solvable 4.2 Showing that the group is metacyclic 5 References 5.1 Textbook references

Statement

Suppose G is a Z-group. In other words, G is a finite group with the property that every Sylow subgroup of G is cyclic, i.e., is a cyclic Sylow subgroup. Then, G is a metacyclic group: it has a cyclic normal subgroup such that the quotient is also a cyclic group.

Further, we can choose the cyclic normal subgroup so that the quotient acts faithfully on it, and hence, the quotient is a subgroup of the automorphism group of the original cyclic normal subgroup.

Related facts

• Classification of cyclicity-forcing numbers
• Coprime automorphism group implies cyclic with order a cyclicity-forcing number

Applications

• Square-free implies solvability-forcing

Facts used

1. Cyclic Sylow subgroup for least prime divisor has normal complement
2. Solvable implies Fitting subgroup is self-centralizing
3. Solvability is extension-closed: An extension of a solvable normal subgroup by a solvable quotient group is solvable.
4. Quotient group acts on abelian normal subgroup
5. Cyclic implies aut-abelian

Proof using advanced methods

Showing that the group is solvable

We induct on the order.

Given: A finite group G of order where p_i are distinct primes. Every Sylow subgroup of G is cyclic.

To prove: G is solvable.

Proof: Without loss of generality, assume that p₁ is the least prime divisor of n. Let P₁ be a p₁-Sylow subgroup.

1. Fact (1) tells us that P₁ has a normal complement N in G.
2. Every Sylow subgroup of N is cyclic: N has order , so any Sylow subgroup of N is Sylow in G, hence cyclic. Thus, we can apply the induction hypothesis, and obtain that N is solvable.
3. G is solvable: G has a solvable normal subgroup N and a quotient G / N that is a p₁-group, hence nilpotent (in fact, an application of the second isomorphism theorem yields that the quotient is isomorphic to P₁, which is cyclic). In particular, both N and G / N are solvable, so G is solvable by fact (3).

Showing that the group is metacyclic

Given: A solvable finite group G of order where p_i are distinct primes. Every Sylow subgroup of G is cyclic.

To prove: G is metacyclic, with a cyclic normal subgroup and a cyclic quotient acting faithfully on it by conjugation.

Proof: Let F(G) be the Fitting subgroup of G.

1. : This follows from fact (2).
2. F(G) is cyclic: First note that F(G) is nilpotent, so it is a direct product of its Sylow subgroups. Any Sylow subgroup of F(G) is a p-subgroup of G, hence is contained in a p-Sylow subgroup. Thus, it is cyclic. This yields that F(G) is cyclic.
3. G / F(G) acts faithfully on F(G) by conjugation: Since F(G) is cyclic, it is in particular abelian, so we get a well-defined action of the quotient group G / F(G) on F(G) (fact (4)). The kernel of this is C_G(F(G)) / F(G), but as we concluded in step (1), this is trivial. Thus, we have a faithful action of G / F(G) on F(G).
4. G / F(G) is cyclic: Since F(G) is cyclic, its automorphism group is abelian, and from the previous step, G / F(G) is isomorphic to a subgroup of . Thus, G / F(G) is abelian. On the other hand, all its Sylow subgroups are cyclic, since these are quotients of Sylow subgroups of G, which are cyclic. Thus, G / F(G) is itself cyclic.

The conclusions of steps (2), (3), and (4) complete the proof.

References

Textbook references

• The Theory of Groups by Marshall Hall, Jr., Page 146, Theorem 9.4.3, (Proof uses counting arguments and is about two pages long.)^{More info}
• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 258, Theorem 6.2, Chapter 7 (Fusion, transfer and p-factor groups), Section 7.6 (elementary applications), ^{More info}