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 groupsXifw(G) is finite wheneverGwis finite for a groupG∈χ. It is a...
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 ...
Words You Always Have to Look Up How to Use Em Dashes (—), En Dashes (–) , and Hyphens (-) Words in Disguise: Do these seem familiar? Why is '-ed' sometimes pronounced at the end of a word? Democracy or Republic: What's the difference?
We prove that if G is a finite group, N is a normal subgroup, and there is a prime p so that all the elements in G ∖ N have p-power order, then either G is a p-group or G = PN where P is a Sylow p-subgroup and (G,P,P∩ N) is a Frobenius–Wielandt triple. We al...
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 ...
Let G be a group of order pq, where p,q are two distinct primes. If G has exactly one subgroup of order p and exactly one subgroup of order q. In the following statements, ( ) is certainly true. A G is non-Abelian B G is Abelian but not cyclic C G is cyclic D |Z(G)|...
1) order of an element to a subgroup 元素对于子群的阶 1. In this paper,the author discusses the order of an element of periodic groups and free 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...
1) The Order of a Fuzzy Subgroup 模糊子群的阶 2) Step normal fuzzy subgroup 阶梯模糊子群 3) fuzzy subgroups 模糊子群 1. A note on fuzzy relations andfuzzy subgroups; 关于模糊关系与模糊子群的注记 2. we give definition of normal α+β-fuzzy subgroupsand discuss some properties of α+β-fu...
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,...