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

Where G is a group, and a, b \in G, it is always the case that

|ab| = |ba|.

That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.

Advertisements

106: Group Isomorphisms

In group theory, a group isomorphism between 2 groups G, H does not simply refer to an ‘isomorphism’ between the underlying sets of G, H, 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 \varphi : G \mapsto H is a group isomorphism, then

  • |G| = |H|
  • G is abelian \leftrightarrow H is abelian
  • for all x \in G, |x| = |\varphi (x)|.

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

Let G be a group and x be any element of that group. Then the functions

f: a \mapsto xa

g: a \mapsto ax

are bijective.

Proof:

Consider the function f. By the left-cancellation property of groups, if ab = ac, then b = c. This means that if b \neq c, then ab \neq ac, and it follows that f is injective. Since the domain and codomain of f are the same (G), this means that f is in fact bijective.

The proof for bijectivity of g 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

In general, the symmetric group S_n (the set of permutations of n distinct objects under permutation composition) is non-abelian. The somewhat trivial exception is when n \leq 2.

92: Order (Groups)

The order of a group G, often denoted |G|, is the cardinality of its underlying set. If the order is finite, then G is called a finite group, and likewise for infinite order.

An example of a group with finite order would be the dihedral group D_n for some n. An example of one with infinite order would be the group of \mathbb{Z} together with addition.

85: The Group as a Category

The group can be viewed as equivalent to a specific kind of category, specifically the category C 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 C, the group operation to \circ (composition), and the group identity to the identity morphism on the single element of C. Since all morphisms in C are isomorphisms, each ‘element’ has an inverse, and \circ is associative, analogous to the associativity of the group operation.