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# Section2.5Group conventions and properties¶ permalink

Before we discuss more examples, we present a theorem and look at some conventions we follow and notation we use when discussing groups in general; we also discuss some properties of groups.

# Subsection2.5.1Some group conventions

##### Convention2.5.2

We will generally use $e$ or $e_G$ as our default notation for an identity element of group, but be aware that many mathematicians denote a group's identity element by 1. We denote the inverse of element $a$ in $G$ by $a^{-1}\text{.}$

##### Warning2.5.3

Although it is written in what we call multiplicative notation, do not assume $a^{-1}$ is what we usually think of as a multiplicative inverse for $a\text{;}$ remember, we don't even know if elements of a group are numbers! The type of inverse that $a^{-1}$ is (a multiplicative inverse for a real number? an additive inverse for a real number? a multiplicative inverse for a matrix? an inverse function for a function from $\R$ to $\R\text{?}$) depends on both $G$'s elements and its operation.

##### Convention2.5.4

• We usually don't use the notation $*$ when describing group operations. Instead, we use the multiplication symbol $\cdot$ for the operation in an arbitrary group, and call applying the operation “multiplying”—even though the operation may not be “multiplication” in the non-abstract, traditional sense! It may actually be addition of real numbers, composition of functions, etc. Moreover, when actually operating in a group $\langle G, \cdot\,\rangle\text{,}$ we typically omit the $\cdot$ . That is, for $a,b\in G\text{,}$ we write the product $a\cdot b$ as $ab\text{.}$ We call this the “product” of $a$ and $b\text{.}$

For every element $a$ in a group $\Gdot$ and $n\in \Z^+\text{,}$ we use the expression $a^n$ to denote the product

\begin{equation*} a \cdot a \cdot \cdots \cdot a \end{equation*}

of $n$ copies of $a\text{,}$ and $a^{-n}$ to denote $(a^{-1})^n$ (that is, the product of $n$ copies of $a^{-1}$). Finally, we define $a^0$ to be $e\text{.}$ Note that our “usual” rules for exponents then hold in an arbitrary group: that is, if $a$ is in group $\langle G, \cdot\,\rangle$ and $m,n\in \Z\text{,}$ then $a^m a^n = a^{m+n}$ and $(a^m)^n=a^{mn}=(a^n)^m\text{.}$

However:

• When we know our operation is commutative, we typically use additive notation, denoting the group operation by $+\text{,}$ calling the group operation “addition,” and denoting the inverse of an element $a$ by $-a\text{.}$ When we use additive notation, we do not omit the $+$ when operating in a group $\langle G,+\rangle\text{,}$ and we call $a+b$ a sum rather than a product. Also, when working with an operation that is known to be commutative, the identity element may be denoted by 0 rather than by $e\text{,}$ $e_G\text{,}$ or 1, and for $n\in \N\text{,}$ we write $na$ instead of $a^n\text{.}$

Finally, note that $(-n)a=n(-a)=-(na)$ (where $-a$ and $-(na)$ indicate the additive inverses of $a$ and $na\text{,}$ respectively); we can therefore unambiguously use the notation $-na$ for this element. Using this notation, note that for $m\in \Z\text{,}$ $na+ma=(n+m)a$ and $n(ma)=(nm)a\text{.}$

##### Warning2.5.5

Be careful to always know where an element you are working with lives! For instance, if, as above, $n\in \Z^+$ and $a$ is a group element, $-n$ and $-a$ look similar but may mean very different things. While $-n$ is a negative integer, $-a$ may be the additive inverse of a matrix in $\M_2(\R)\text{,}$ the additive inverse 2 of the number 4 in $\Z_6\text{,}$ or even something completely unrelated to numbers.

##### Remark2.5.6

Multiplicative notation can be used when working with any arbitary group, while additive notation should be used only when working with a group whose binary operation is commutative.

We summarize multiplicative versus addition notation in the following table, where $a,b$ are elements of a group $G\text{.}$

##### Convention2.5.8

We do use the notation $*$ when using multiplicative or additive notation would lead to confusion. For instance, if we want to define an operation on $\Q^*$ that assigns to pair $(a,b)$ the quantity $ab/2\text{,}$ it would be unwise to use multiplicative or additive notation for this operation since we already have conventional meanings of $ab$ and $a+b\text{.}$ Similarly, we would not denote the identity element of $\Q^*$ under this operation by $0$ or $1\text{,}$ since the identity element in this group is the rational number $2\text{,}$ and writing $0=2$ or $1=2$ would look weird.

Finally:

##### Convention2.5.9

If there is a default notation for a particular operation (say, $\circ$ for composition of functions) or identity element (say, $I_n$ in $GL(n,\R)$) we usually use that notation instead.

# Subsection2.5.2Some group properties

While we don't need to worry about “order” when multiplying a group element $a$ by itself, we do need to worry about it in general.

##### Warning2.5.10

Group operations need not be commutative!

##### Definition2.5.11

A group $\langle G, \cdot\,\rangle$ is said to be abelian if $ab=ba$ for all $a,b\in G\text{.}$ Otherwise, $G$ is nonabelian. (The word “abelian” derives from the surname of mathematician Niels Henrik Abel.)

##### Remark2.5.12

If we know that a binary operation $\cdot$ on a set $G$ is commutative, then in checking to see if axioms $\G_2$ and $\G_3$ hold we need only verify that there exists $e\in G$ such that $ae=a$ (we don't need to check that $ea=a$) for all $a\in G$ and that for each $a\in G$ there exists $b\in G$ such that $ab=e$ (we don't need to check that $ba=e$).

##### Remark2.5.13

If $G$ is not known to be abelian, we must be careful when multiplying elements of $G$ by one another: multiplying on the left is, in general, not the same as multiplying on the right!

##### Definition2.5.14

If $G$ is a group, then the cardinality $|G|$ of $G$ is called the order of $G$. If $|G|$ is finite, then $G$ is said to be a finite group; otherwise, it's an infinite group.

##### Example2.5.15

Of the groups we've discussed, which are abelian? Which are infinite/finite?

We have already seen that identity elements of groups are unique, and that each element $a$ of a group $G$ has a unique inverse $a^{-1}\in G\text{.}$ Here are some other basic properties of groups.

##### Warning2.5.18

We only of necessity have $(ab)^{-1}=a^{-1}b^{-1}$ if $G$ is known to be abelian!

However, we do have the following: