Letwbe a group-word. Given a groupG, we denote byw(G) the verbal subgroup corresponding to the wordw, that is, the subgroup generated by the setGwof allw-values inG. The wordwis called concise in a class of gro
How to tell if a group is cyclic? How to tell if two groups are isomorphic? How do you order 1/3, 0.3, 25%, and 2/5 in ascending order? Prove that every group of order 30 has a normal subgroup of order 3 or 5. Find the order of the element R_{270} in the group D_4. ...
Prove that there can be one nontrivial homomorphism from S_3 \to Z_3. Hint: What are the normal subgroup of How to show if the group is a trivial subgroup? Find the order of the cyclic subgroup of Z_{4} generated by 3. Show that, in an Abelian group G, the set ...
If G is a p-group of order pn such that |M(G)| attains the bound, then for every central subgroup K of order p, |M(G/K)| also attains the bound. Lemma 3.3 There is no group G of order pn (n≥4) having maximal class such that |M(G)| attains the bound. Proof First we ...
No group of order36 is simple.Such a group G has either one or four subgroups of order 9.If there is only one such subgroup,it is normal in G.If there are four such subgroups,let H and K be two of them.H∩K must have at least 3elements,or Hk would have to have 81elements,...
Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic gr
class 3 has been classified. Furthermore, Hu et al. [19] classified regular 2-maps for maximal class by using the classification of 2-groups with a cyclic maximal subgroup. There are many papers [11,16,23,27,28] showing that for any given positive integersmandlsatisfying...
Finite group Subgroups of non-prime-power order TI-subgroups Frobenius groups 1. Introduction Throughout this paper all groups are finite. Let G be a group and R be a subgroup of G. If for each g∈G we always have R∩Rg=1 or R then we say that R is a TI-subgroup of G. The ...
groups,studies the order of an element of the additive group of a ring and establishes the concept of the characteristic number of a ring,and gives two popularizations of the order of an element of groups:the order of an element to a subgroup and the order of an element of semigroups. ...
(a) Draw a Cayley graph for the group G wit h generators a and b with a~5=b^4=eandb^(-1)ab=a~2(b) Show that the subgroup H generated by a and b^2 is isomorphi c to the dihedral group D5 of order 10.(c) Find an element c in G of order 4 such t hat b and c ...