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On the number of subgroups of given order in a finite $ p$-group of exponent $ p$


Author: Ya. G. Berkovich
Journal: Proc. Amer. Math. Soc. 109 (1990), 875-879
MSC: Primary 20D15
DOI: https://doi.org/10.1090/S0002-9939-1990-1017844-1
MathSciNet review: 1017844
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Abstract: A. Kulakoff [1] showed that a noncyclic $ p$-group of order $ {p^m},p > 2$, contains $ 1 + p + k{p^2}$ subgroups of order $ {p^n},0 < n < m$, where $ k$ is a nonnegative integer. In this note we show that for $ 1 < n < m - 1$ a $ p$-group of order $ {p^m}$ and exponent $ p$ contains $ 1 + p + 2{p^2} + k{p^3}$ subgroups of order $ {p^n}$.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1017844-1
Article copyright: © Copyright 1990 American Mathematical Society