Today I Learned

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

Category Archives: abstract algebra

117: Generality of the 2-Element Boolean Algebra

The behavior of boolean algebras in general is captured by that of 2, the boolean algebra with 2 elements, since a theorem holds for 2 if and only if it also holds for any boolean algebra.

108: Generating Set for the Symmetric Group

There are many possible choices of a generating set for S_n, the symmetric group. However, one particularly simple one is the set

\{(0, 1, 2, \dots, n-1), (0, 1)\}

where 0, 1, 2, \dots, n-1 is an ordering of the elements being permutated by the elements of S_n, (0, 1, 2, \dots, n-1) is the permutation ‘shifting’ all these elements to their successor, and (0, 1) 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 n elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the n elements to an arbitrary position in the ordering.

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.

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)|.

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.

81: General Linear Groups

The general linear group \textrm{GL}(V), where V is a vector space, is the group of all automorphisms T: V \rightarrow V under the operation of composition of linear transformations.

When V is over the field F and is finite-dimensional with \textrm{dim}(V) = n, this group is isomorphic to the group of invertible n \times n matrices with entries from F under the operation of matrix multiplication. In this case, the group is often written as \textrm{GL}(n, F).

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

f: G \rightarrow \textrm{GL}(V)

where G is the group being represented and V is any vector space.

80: Multilinear Forms

In abstract algebra, a multilinear form is a mapping

f: V^n \rightarrow F

where V is a vector space over the field F, such that each argument of f is linear over F with the other arguments held fixed. A special case of this is when n = 2 and f is a bilinear form.