搜尋結果
2016年2月13日 · 7. Rotman's "Introduction to the Theory of Groups" is a great book. It is focused on groups only (unlike some books on abstract algebra that sometimes skim over the subject), and Rotman's style makes it very readable. He usually includes proofs to every claim, a good deal of examples, and useful exercises. Share.
2017年6月1日 · 10. Z ×Z Z × Z is a not a free group, but Z ∗Z Z ∗ Z is! – Dietrich Burde. Jun 1, 2017 at 11:38. "Free" means that relations never hold unless they have to (from the group axioms). Powers of one element have to commute among each other, but beyond that elements do not have to commute; hence free groups on more than one generator cannot ...
2016年10月17日 · Conjugacy is one of the more important concepts in group theory. It's an equivalence relation, an easy exercise. The class equation gives, for a finite group, the sizes of the various conjugacy classes. This equation is very useful. For instance you can read off the size of the center (the sum of the ones, since elements of the center are in ...
2011年11月27日 · Basically, there is a theorem, called the Jordan–Hölder theorem, which says that simple groups are the basic building blocks of group theory. Every group is, in some senses, composed of these groups. So if you know what the finite simple groups are, then you "know" all other finite groups, for a suitable value of "know". – user1729.
A component of G is a subnormal subgroup L such that L = [L, L] and L / Z(L) is simple. It turns out that distinct components centralize each other. The group E(G) is the central product of all the components of G (and G permutes its components by conjugation).
19. That operation on cosets is well-defined if and only if H is a normal subgroup. If H is just a subgroup, what you call "left quotient group" has the more standard name "set with a left group action". More precisely, the coset spaces G/H describe essentially all the examples of sets with transitive left G-actions.
Discriminants Define β = ∏i <j(θi − θj) and Δ = β2. Since Δ is symmetric in the θ 's, it is in the ground field K and is quite reasonable to compute. The number Δ is called the discriminant of f. The Galois group G is contained in An if and only if β is in K; in other words, if and only if Δ is square in K.
The order of an element g of a group G is the smallest positive integer n: gn = e, the identity element. I understand how to find the order of an element in a group when the group has something to with modulo, for example, in the group. U(15) = the set of all positive integers less than n and relatively prime to n.
2020年7月23日 · In case n = 2 n = 2 Obviously, we have that the fundamental group of B B is 1 1 and of A A is Z Z and consider the element of fundamental groub of A A which is a a. We have that a2 a 2 is in the intersection of A A and B B. Thus, a2 a 2 shoud be 1 1 in fundamental group of RP2 R P 2. Therefore the fundamental group of RP2 R P 2 shoud be Z/2 Z / 2.
2015年4月10日 · Easy examples: Groups of exponent $1$ are trivial. Groups of exponent $2$ are abelian (this is a standard exercise). Groups of exponent $3$ are not necessarily abelian, as the Heisenberg group over $\mathbb{F}_3$ shows. Burnside's problem is to
