Journal of Prime Research in Mathematics

# On some characterization of nearly Hall S-semiembedded subgroups of finite groups

Iftikhar Ali$$^a$$, Abid Mahboob$$^{b,*}$$, Taswer Hussaain$$^c$$, Faryal Chaudhry$$^a$$
$$^a$$Department of Mathematics and Statistics, The University of Lahore, Pakistan.
$$^b$$Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan.
$$^c$$Department of Mathematics, Punjab Education Department, Lahore, Pakistan.
Correspondence should be addressed to: Abid Mahboob at abid.mahboob@ue.edu.pk

### Abstract

Let K be a subgroup. Then K is known as partially Hall s-semiembedded subgruop in $$\Game$$ if for K$$\mathcal{T}$$ is s-semi permutable in $$\Game$$ and K $$\cap \mathcal{T} \leq$$ $$K_{\tilde{s} \Game}$$ where $$K _{\tilde{s} \Game}$$ generated by all those subgroups of K which are Hall s-semiembedded in $$\Game$$, there exists a normal subgroup $$\mathcal{T}$$ of G. In this paper, we investigate the notion of partially Hall $$S$$-semi embedded subgroups on the structure of finite group $$\Game$$. We obtain some new criteria related to the $$p$$-nilpotency and super solubility of a finite group. Some earlier results about formations are also generalized by our results.

#### Keywords:

Hall s-semiembedded subgroups, Partially $$S$$-embeded subgroup, $$p$$-nilpotent.