Def/Group

Logical Connections

This definition logically relies on the following definitions and statements: Def/Binary operation, Def/Associative, Def/Composite function

The following statements and definitions logically rely on the material of this page: Def/Abelian, Def/Center (group), Def/Conjugate, Def/Coset, Def/Cyclic, Def/Fixed point, Def/Group homomorphism, Def/Kernel (group), Def/Normal subgroup, Def/Order of a group element, Def/P-group, Def/Ring, Def/Sylow subgroup, Def/Unital, Def/Vector space, State/Cancellation in groups, State/Existence of Sylow subgroups, State/First isomorphism theorem for groups, State/Groups of prime order are cyclic, State/Groups of prime squared order are abelian, State/Homomorphic images of subgroups are subgroups, State/Inversion is an antiautomorphism, State/Kernel criterion for injectivity of group homomorphisms, State/Lagranges theorem, State/Uniqueness of identity element in a monoid, State/Uniqueness of inverse elements in a monoid, and Struct/The group with two elements.

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Composition Notation

The binary operation symbol "$\circ$" is sometimes used, especially when the operation arises from composition of functions. In this context, the notation $g^{-1}$ is used for the inverse element of $g$. In this situation, the identity element is usually denoted by $1$, $e$, or by $Id$.

When composition notation is used, repeated composition is denoted with natural number exponents. Namely, if $n \in \NN$, and $g \in G$, one writes: $$g^n = g \circ \cdots \circ g \mbox{ where $g$ appears $n$ times }.$$ More rigorously, the expression $g^n$ is defined recursively:

• When $n = 0$, one defines $g^0$ to be the identity element.
• When $n > 0$, one defines $g^n = g^{n-1} \circ g$.

Since the inverse of an element $g$ is denoted by $g^{-1}$, one uses integer exponents to denote repeated composition of elements or their inverses. In particular, when $n \in \NN$, one writes: $$g^{-n} = (g^{-1}) \circ \cdots \circ (g^{-1}) \mbox{ where $g^{-1}$ appears $n$ times }.$$

In this setting, some common rules for exponents hold:

• For all $a,b \in \ZZ$, and all $g \in G$, $g^{a + b} = g^a \circ g^b$.
• For all $a,b \in \ZZ$, and all $g \in G$, $g^{ab} = (g^a)^b$.

However, unless $G$ is abelian, the following rule does not usually hold:

• For all $g,h \in G$, and all $a \in \ZZ$, $(gh)^a = g^a h^a$.

Multiplicative Notation

Alternatively, the symbol "$\cdot$" is often used in the context of abstract groups, when the operation arises from multiplication in a ring. When the symbol "$\cdot$" is used for the operation in a group, the symbol $1$ is almost always used for the identity element in the group. Moreover, one often writes $gh$ instead of $g \cdot h$ in these situations. One also writes $g^{-1}$ for the inverse element of $g$.

When multiplicative notation is used, integer exponents are also used, as described above.

Additive Notation

In abelian groups, especially those arising from addition in a ring, the symbol "$+$" is sometimes used for the group operation. In this case, the symbol $0$ is used for the identity element in the group. When $g$ is an element of a group, and additive notation is used, one writes $-g$ for the inverse element of $g$. This is consistent with the traditional notation $g + (-g) = 0$.

When additive notation is used, repeated composition is denoted with multiplication by integers. Namely, when $n \in \NN$, one writes: $$n \cdot g = g + \cdots + g \mbox{ where $g$ appears $n$ times }.$$ Of course, $0 \cdot g = 0$. When $n \in \NN$, one also writes: $$-n \cdot g = (-g) + \cdots + (-g) \mbox{ where $-g$ appears $n$ times }.$$ This defines multiplication $n \cdot g$, for all $n \in \ZZ$, and all $g \in G$.

In this setting, the familiar distributive laws, and associative law hold, in the following senses:

• For all $a,b \in \ZZ$, $(a + b) \cdot g = a \cdot g + b \cdot g$.
• For all $a \in \ZZ$, and all $g_1, g_2 \in G$, $a \cdot (g_1 + g_2) = a \cdot g_1 + a \cdot g_2$.
• For all $a,b \in \ZZ$, $ab \cdot g = a \cdot (b \cdot g)$.

The following are examples of groups:

This article has been written by: User:Marty