A characterization of the Mathieu group M12 is established; the characterization is used by Aschbacher and Smith in their classification of the quasithin finite simple groups. This is one in a series of papers providing simplified, modern, computer free treatments of the existence and uniqueness of the sporadic groups, and of the normalizers of subgroups of prime order and Sylow subgroups in th...

We present a method of producing normal subgroups of the free group on r generators not containing a given word. Special attention is given to the normal subgroups of the free group on two generators. In considering the exponent sum of the generators in a given word, we produce such subgroups and clearly state for what class of words these subgroups are not easily obtainable. We also provide a ...

Now that the classification of finite simple groups is complete, it is logical to look at the extension problem. An important special case to consider is when M is a minimal normal subgroup of G and both G/M and M are known groups. If M is abelian, various techniques have been used to derive information about G. Indeed, almost the entire theory of finite solvable groups can be said to rest upon...

Our goal will be to generalize the construction of the group Z/nZ. The idea there was to start with the group Z and the subgroup nZ = 〈n〉, where n ∈ N, and to construct a set Z/nZ which then turned out to be a group (under addition) as well. (There are two binary operations + and · on Z, but Z is just a group under addition. Thus, the fact that we can also define multiplication on Z/nZ will not...

In this paper a first step in classifying the fuzzy normal subgroups of a finite group is made. Explicit formulas for the number of distinct fuzzy normal subgroups are obtained in the particular cases of symmetric groups and dihedral groups.