WebFor S 10 we have that the maximal order of an element consists of 3 cycles of length 2,3, and 5 (or so I think) resulting in an element order of lcm ( 2, 3, 5) = 30. I'm certain that the all of the magnitudes will have to be relatively prime to achieve the greatest lcm, but other than this, I don't know how to proceed. Any thoughts or references? WebIn mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p -group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element.
Group Theory — Order of an Element in the group, generator
WebThe order of an element g in some group is the least positive integer n such that g n = 1 (the identity of the group), if any such n exists. If there is no such n, then the order of g is … In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the … See more The symmetric group S3 has the following multiplication table. • e s t u v w e e s t u v w s s e v w t u t t u e s w v u u t w v e s v v w s e u t w w v u t s e This group has six … See more Suppose G is a finite group of order n, and d is a divisor of n. The number of order d elements in G is a multiple of φ(d) (possibly zero), … See more An important result about orders is the class equation; it relates the order of a finite group G to the order of its center Z(G) and the sizes of its non-trivial conjugacy classes: $${\displaystyle G = Z(G) +\sum _{i}d_{i}\;}$$ See more The order of a group G and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the factorization of G , the more complicated the structure of G. For G = 1, the … See more Group homomorphisms tend to reduce the orders of elements: if f: G → H is a homomorphism, and a is an element of G of finite order, then ord(f(a)) divides ord(a). If f is See more • Torsion subgroup See more 1. ^ Conrad, Keith. "Proof of Cauchy's Theorem" (PDF). Retrieved May 14, 2011. {{cite journal}}: Cite journal requires journal= (help) 2. ^ Conrad, Keith. "Consequences of Cauchy's Theorem" See more inbotna diabetic medication
Maximal order of an element in a symmetric group
WebOct 3, 2016 · 1. Find a group G that contains elements a and b such that a 2 = e, b 2 = e, but the order of the element a b is infinite. My attempt: Clearly G cannot be abelian. So I looked at two commonly known non-abelian groups, namely. (i) The group of symmetries of the equilateral triangle. (ii) 2 by 2 matrices. Neither of these seem to work. WebMar 24, 2024 · In general, finding the order of the element of a group is at least as hard as factoring (Meijer 1996). However, the problem becomes significantly easier if and the … WebIn group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C n, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly … incident of plagiarism