Some of the things I've learned every day since Oct 10, 2016
Monthly Archives: February 2017
February 3, 2017Posted by on
There are many possible choices of a generating set for , the symmetric group. However, one particularly simple one is the set
where is an ordering of the elements being permutated by the elements of , is the permutation ‘shifting’ all these elements to their successor, and is the permutation swapping the first 2 of these.
I won’t provide a formal proof here as it’s a little tedious, but you can easily convince yourself that every permutation of these elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the elements to an arbitrary position in the ordering.
February 2, 2017Posted by on
Where is a group, and , it is always the case that
That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.
February 1, 2017Posted by on
In group theory, a group isomorphism between 2 groups does not simply refer to an ‘isomorphism’ between the underlying sets of , but to a special type of homomorphism between them: one which is bijective. In a sense, isomorphic groups are the same group, just with different symbols representing their elements and operations — something which is not necessarily true if they’re homomorphic or if just their underlying sets are isomorphic.
In particular, if is a group isomorphism, then
- is abelian is abelian
- for all .