All groups considered are finite. In definitions and notation we follow [1].
A normally hereditary class of groups \(\frak{F}\) is called a Fitting class if it is closed under the products of normal \(\frak{F}\)-subgroups.
If \(\emptyset\neq\frak{F}\) is a Fitting class then a subgroup \(G_\frak{F}\) of a group \(G\) is called its \(\frak{F}\)-radical if it is the largest normal \(\frak{F}\)-subgroup of \(G\).
A nonempty Fitting class \(\frak{F}\) is called normal in a Fitting class \(\frak{X}\) or \(\frak{X}\)-normal (this is denoted by \(\frak{F}\unlhd\frak{X}\))
if \(\frak{F}\subseteq\frak{X}\) and for every \(\frak{X}\)-group \(G\) a subgroup \(G_\frak{F}\) is \(\frak{F}\)-maximal in \(G\).
Recall that the product of Fitting classes \(\frak{X}\) and \(\frak{Y}\) is the class of groups \(\frak{XY}=(G:~G/G_\frak{X}\in\frak{Y})\) which is also a Fitting class.
In the class \(\frak{S}\) of all soluble groups, it is established [2] that the product of Fitting classes \(\frak{F}\) and \(\frak{H}\) is \(\frak{S}\)-normal when at least one factor is normal in \(\frak{S}\).
Since the operation of multiplication of Fitting classes is associative, this fact signifies that the algebra \(<\cal{P},\cdot>\) is a semigroup (\(\cal{P}\) is the set of all \(\frak{S}\)-normal Fitting classes and "\(\cdot\)" is the operation of multiplication of Fitting classes).
Later, a fact similar to the one mentioned above was proved [3] for the case of \(\frak{E}\)-normal Fitting classes where \(\frak{E}\) denotes the class of all groups.
Note that for the case of \(\frak{S}_\pi\)-normal Fitting classes (\(\frak{S}_\pi\) is the class of all soluble \(\pi\)-groups) the result [2] was extended in [4].
This leads us to the question of whether such property holds in passing from soluble groups to arbitrary ones.
Let \(\mathbb{P}\) be the set of all primes, \(\emptyset\neq\pi\subseteq\mathbb{P}\), and let \(\frak{E}_\pi\) be the class of all \(\pi\)-groups.
If a Fitting class \(\frak{F}\) is normal in \(\frak{E}_\pi\) then we call it \(\pi\)-normal.
The following theorem gives a positive answer to the question of whether the algebra \(<\cal{P},\cdot>\) is a semigroup, where \(\cal{P}\) is the set of all \(\pi\)-normal Fitting classes.
Theorem. Let \(\frak{X}\) and \(\frak{Y}\) be Fitting classes such that \(\frak{X}\subseteq\frak{E}_\pi\) and \(\frak{Y}\unlhd\frak{E}_\pi\).
Then \(\frak{XY}\unlhd\frak{E}_\pi\).
In the case \(\pi=\mathbb{P}\), the theorem implies the result of Laue [3, Theorem 2.7].
References
K. Doerk, T. Hawkes, Finite soluble groups, Berlin–New York: Walter de Gruyter (1992), 891 p.
J. Cossey, Products of Fitting Classes, Math. Z., 141 (1975), 289–295.
H. Laue, Über nichtaflösbare normale Fittingklassen, J. Algebra, 45 (1977), 274–283.
N. V. Savelyeva, Maximal subclasses of \(\pi\)-normal Fitting classes, Herald of Polotsk State University, Series C. Fundamental Sciences, N9 (2008), 22–31 (in Russian).
