#### Glasnik Matematicki, Vol. 45, No.2 (2010), 431-439.

### ALTERNATE PROOFS OF TWO CLASSICAL THEOREMS ON FINITE SOLVABLE GROUPS AND SOME RELATED RESULTS FOR *P*-GROUPS

### Yakov Berkovich

Department of Mathematics,
University of Haifa,
Mount Carmel, Haifa 31905,
Israel

**Abstract.** We offer a new proof of the classical theorem asserting that if a positive
integer *n* divides the order of a solvable group *G* and the set *L*_{n}
of solutions of the equation *x*^{n}=1 in *G* has cardinality *n*,
then *L*_{n} is a subgroup of *G*. The second proof of that theorem is also presented.
Next we offer an easy proof of Philip Hall's theorem on solvable groups independent of Schur-Zassenhaus' theorem.
In conclusion, we consider some related questions for *p*-groups. For example, we study the irregular
*p*-groups *G* satisfying *|L*_{pk}| ≤ p^{k+p-1} for *k* > 1.

**2000 Mathematics Subject Classification.**
20D15.

**Key words and phrases.** Solvable groups, Philip Hall's theorem on solvable groups, irregular *p*-groups, *p*-groups of maximal class.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.45.2.10

**References:**

- Y. Berkovich,
*Alternate proofs
of some basic theorems of finite group theory*, Glas. Mat. Ser. III **40(60)** (2005), 207-233.

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CrossRef

- Y. Berkovich, Groups of Prime Power Order, Volume 1, Walter de Gruyter, Berlin, 2008.

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- Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, Walter de Gruyter, Berlin, 2008.

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- Y. G. Berkovich and E. M. Zhmud, Characters of Finite Groups. Part 1, Translations of Mathematical Monographs, Volume
**172**, American Mathematical Society, Providence, 1998.

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- M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959.
%nije referirano

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- Philip Hall,
*A note on solvable groups*, J. London Math. Soc. **3** (1928), 98-105.

- N. Iivory and H. Yamaki,
*On a conjecture of Frobenius*, Bull. Amer. Math. Soc. (N.S.) **25** (1991), 413-416.

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CrossRef

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