# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## Category Archives: group theory

## 107: The Order of the Product of 2 Group Elements

February 2, 2017

Posted 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.

## 106: Group Isomorphisms

February 1, 2017

Posted 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 .

## 104: Bijectivity of Left- or Right-Multiplication by Group Elements

January 30, 2017

Posted by on Let be a group and be any element of that group. Then the functions

are bijective.

#### Proof:

Consider the function . By the left-cancellation property of groups, if , then . This means that if , then , and it follows that is injective. Since the domain and codomain of are the same (), this means that is in fact bijective.

The proof for bijectivity of is identical, but uses the right-cancellation property instead.

The intuitive significance of this is that left- or right-multiplication by a given element always sends distinct elements to distinct elements, which is pretty neat.

## 101: Non-Commutativity of Symmetric Groups

January 26, 2017

Posted 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 .

## 92: Order (Groups)

January 16, 2017

Posted 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.

## 85: The Group as a Category

January 5, 2017

Posted by on The group can be viewed as equivalent to a specific kind of category, specifically the category with only a single element and all of whose morphisms are isomorphisms. In this equivalence, the elements of the group correspond to the morphisms of , the group operation to (composition), and the group identity to the identity morphism on the single element of . Since all morphisms in are isomorphisms, each ‘element’ has an inverse, and is associative, analogous to the associativity of the group operation.

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