Some of the things I've learned every day since Oct 10, 2016
Category Archives: abstract algebra
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 .
January 26, 2017Posted by on
In general, the symmetric group (the set of permutations of distinct objects under permutation composition) is non-abelian. The somewhat trivial exception is when .
January 16, 2017Posted by on
The order of a group , often denoted , is the cardinality of its underlying set. If the order is finite, then is called a finite group, and likewise for infinite order.
An example of a group with finite order would be the dihedral group for some . An example of one with infinite order would be the group of together with addition.
January 1, 2017Posted by on
The general linear group , where is a vector space, is the group of all automorphisms under the operation of composition of linear transformations.
When is over the field and is finite-dimensional with , this group is isomorphic to the group of invertible matrices with entries from under the operation of matrix multiplication. In this case, the group is often written as .
General linear groups are used in group representations. A group representation is a representation of a group as a general linear group. That is, it is an automorphism
where is the group being represented and is any vector space.
December 31, 2016Posted by on
In abstract algebra, a multilinear form is a mapping
where is a vector space over the field , such that each argument of is linear over with the other arguments held fixed. A special case of this is when and is a bilinear form.