In this paper, we mainly consider some cardinal invariants and grasps of quasitopological groups and some properties of two classes of quasitopological groups. We show that: (1) There exists a pseudocompact quasitopological group K with countable cellularity and $In(K)>\omega$, which gives a negative answer to [30, Question 5.3] ([27, Question 3.6]); (2) There exists a pseudocompact quasitopological group G of countable cellularity and uncountable g-tightness, which gives a negative answer to [8, Open problem 6.4.9]; (3) There exists a Tychonoff quasitopological group G containing compact invariant subgroups $F,M$ such that $G=FM$, but the space G is not Čech-complete, which gives a partial answer to [8, Open problem 4.6.9]; (4) A Fréchet-Urysohn quasitopological group G with sequentially continuous multiplication is a strong ${\alpha }_4$-space; and as an application, we give a partial answer to [9, Question 2.4]. We also introduce the concept of strong quasitopological groups. Some properties of strong quasitopological groups are obtained, which generalize some properties of topological groups. As some applications, we give some partial answers to related open problems.

在本文中，我们主要考虑一些基本变量和拟拓扑群的掌握以及两类拟拓扑群的一些性质。我们证明：（1）存在一个具有可数细胞数的伪紧凑准拓扑群K和$\mathrm{一世}ñ（ķ）>\omega$，它对[30，问题5.3]（[27，问题3.6]）给出否定的回答；（2）存在一个可数细胞数和不可数g-紧密度的拟紧凑准拓扑群G，它对[8，开放问题6.4.9]给出否定的答案；（3）存在一个包含紧凑不变子群的Tychonoff准拓扑群G$F，\mathrm{中号}$ 这样 $G=F\mathrm{中号}$，但是空间G不是Čech完全的，这部分回答了[8，开放问题4.6.9]；（4）具有连续连续乘法的Fréchet-Urysohn准拓扑群G是一个强${\alpha }_4$-空间; 作为应用，我们对[9，问题2.4]给出部分答案。我们还介绍了强拟拓扑群的概念。获得了强拟拓扑基团的某些性质，这些性质概括了拓扑基团的某些性质。作为某些应用程序，我们对相关的开放性问题给出了部分答案。