### $(*)$-groups and pseudo-bad groups.

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The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^{1}(\mathbb{K}\left(1\right|3),\mathrm{\U0001d52c\U0001d530\U0001d52d}(2,3);{\mathcal{D}}_{\lambda ,\mu}\left({S}^{1|3}\right))$ of the contact Lie superalgebra $\mathbb{K}\left(1\right|3)$ with coefficients in the space of differential operators ${\mathcal{D}}_{\lambda ,\mu}\left({S}^{1|3}\right)$ acting on tensor densities on ${S}^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left({X}_{f}\right)={D}_{1}{D}_{2}{D}_{3}\left(f\right){\alpha}_{3}^{1/2}$, ${X}_{f}\in \mathbb{K}\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{\U0001d52c\U0001d530\U0001d52d}(2,3)$ of $\mathbb{K}\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group...

Let $G$ be a finite group and ${\pi}_{e}\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi}_{e}\left(G\right)$ and ${m}_{k}$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{{m}_{k}:k\in {\pi}_{e}\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups ${L}_{2}\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|{L}_{2}\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, ${p}^{2}-1$, $p(p+\u03f5)/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than...

Let $\omega \left(G\right)$ denote the set of element orders of a finite group $G$. If $H$ is a finite non-abelian simple group and $\omega \left(H\right)=\omega \left(G\right)$ implies $G$ contains a unique non-abelian composition factor isomorphic to $H$, then $G$ is called quasirecognizable by the set of its element orders. In this paper we will prove that the group $PS{L}_{4}\left(5\right)$ is quasirecognizable.