Wikipedia has separate pages for symmetric group and permutation group, but I don't understand what the difference between them is. A symmetric group on a set is the set of all bijections …

Write $\alpha$ and $\beta$ one above the other, so that cycles of the same length coincide; interpret this as a $2$-line description of a permutation, completing it to an element of …

Oct 4, 2016 · I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if $$ a= (1\,3\,5\,2),\quad b= (2\,5\,6 ...

Oct 11, 2020 · You already know the crucial pieces of information: the order of a cycle is its length, and the order of two group elements that commute is the least common multiple of …

Mar 6, 2015 · I understand that the subgroups of order 2 must contain the identity element, and a self inverse element, but I am struggling to find the general form for a self inverse element, I …

Jan 22, 2024 · Answering simply: A permutation group G is transitive over a finite set X if for every x,y in X, there is a g in G such that y=gx, i.e. there is a g that maps x to y.

Aug 20, 2014 · Any group is generated by all of its elements. In particular, if you've listed all the elements, you've got a set of generators. Are there other conditions your generating set must …

Feb 26, 2021 · So am I to assume that cyclic, alternating, lie groups, and sporadic groups are all actually permutation groups? As an additional question, if they all are, why identify finite simple …

Dec 1, 2016 · I imply from carley theorem that every group isomorphism with permutation group

Yep, any group acts on itself in a number of ways. One way is by left/right "multiplication" and another is by conjugation.