International Journal of Group Theory
Volume:10 Issue: 1, Mar 2021

‎In this paper we prove that some Janko groups are uniquely‎ ‎determined by their orders and one irreducible character‎ ‎degree‎. ‎Also we prove that some finite simple $K_4$-groups are‎ ‎uniquely determined by their character degree graphs and their‎ ‎orders‎.

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We calculate the character table of a sharply $5$-transitive subgroup of $alter(12)$‎, ‎and of a sharply $4$-transitive subgroup of $alter(11)$‎. ‎Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups‎, ‎and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations‎.

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It is known that if $ G=AB $ is a product of its totally permutable subgroups $ A $ and $ B $‎, ‎then $ Gin mathfrak{F} $ if and only if $ Ain mathfrak{F} $ and $ Bin mathfrak{F} $ when $ mathfrak{F} $ is a Fischer class containing the class $ mathfrak{U} $ of supersoluble groups‎. ‎We show that this holds when $ G=AB $ is a weakly totally permutable product for a particular Fischer class‎, ‎$ mathfrak{F}diamond mathfrak{N} $‎, ‎where $ mathfrak{F} $ is a Fitting class containing the class $ mathfrak{U} $ and $ mathfrak{N} $ a class of nilpotent groups‎. ‎We also extend some results concerning the $ mathfrak{U} $-hypercentre of a totally permutable product to a weakly totally permutable product‎.

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Let $D$ be a division ring with center $F$ and assume that $N$ is a locally soluble almost subnormal subgroup of the multiplicative group $D^*$ of $D$‎. ‎We prove that if $N$ is algebraic over $F$‎, ‎then $N$ is central‎. ‎This answers partially cite[Conjecture 1]{hai_13}‎.

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‎We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup‎. ‎We show that $|G|$ has at most three prime divisors‎. ‎When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order‎.

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