# Today I Learned

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

## 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.