Tour:Factsheet one (beginners): Difference between revisions

Latest revision as of 23:57, 31 January 2013

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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Here are some facts we have seen so far:

A group can be defined as a set with three operations: a binary operation, called the multiplication or product, an inverse map, and an identity element. These three operations are subject to the conditions of associativity, identity element and inverses.
A group can also be defined as a set with an associative binary operation, called the multiplication or product, such that there exists an identity element and every element has an inverse.
The two definitions of group are equivalent because the identity element is unique, and inverses, when they exist, are unique. We'll see proofs of this in part two.
Subgroups are subsets that are closed under all the group operations. Just having a subset that is closed under the multiplication operation is not enough: for instance, the nonzero integers are closed under multiplication inside the group of nonzero rationals under multiplication, but they don't form a subgroup.
There are two somewhat different ways of defining subgroups: one, where we demand closure under all the group operations, and the other, where we demand closure only under the multiplication, but insist that the subset form a group with this induced operation. The two definitions are equivalent, and we'll see a proof of this in part two.
There is a third way of defining subgroups in terms of left quotients of elements. We'll see, in part two, a proof of the equivalence of this with the other definitions.
A very special class of groups is the abelian groups. A group is abelian iff any two elements commute i.e. the binary operation on it is commutative. Any subgroup of an abelian group is abelian.
The multiplication operation in an abelian group is often denoted additively.
The trivial group is the group with just one element.

Notational conventions

Groups are often denoted by letters like $G, H, K$
The multiplication operation is denoted by $*$ or $\cdot$ , or by omission. Because of associativity of multiplication, we can omit parentheses when multiplying more than two elements, and, when the context is clear, omit the multiplication symbol as well.
The inverse operation is denoted by a superscript of $- 1$ . The superscript applies only to the immediately preceding expression. Thus $x y^{- 1}$ is $x * (y^{- 1})$ and not $(x * y)^{- 1}$
A product of the same element with itself many times is denoted by a power of that element. So $x^{n} = x x x \dots$ $n$ times
The multiplicative identity is denoted by $e$ or $1$
For abelian groups, $+$ denotes the addition, and iterated sum is denoted by integer multiplication. So $n x = x + x + \dots + x$ done $n$ times.
$-$ denotes the additive inverse in an abelian group, and $0$ denotes the additive identity.
Subgroups in general are denoted by the $\leq$ sign. So $H \leq G$ means $H$ is a subgroup of $G$ . We can also say $H \subseteq G$ , but the latter is also used for mere subsets, that aren't subgroups.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
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 {{guided tour|beginners|Introduction one|Entertainment menu one (beginners)|Understanding the definition of a group}}
-Below is a list of some important facts that can be proved directly from the definitions we have seen so far of group, subgroup, trivial group, identity element, Abelian group. We'll see these facts (and more) with full proof, in part two of the guided tour. Links to the full proof statements are provided. To make best use of the tour as a learning experience, it is suggested that you ''try'' to prove these statements yourself.
+Here are some facts we have seen so far:
-# The identity element in a group is unique. This comes from a more general fact: in any [[magma]] (i.e. a set with binary operation), ''if'' there exists a left neutral element (a left identity for the binary operation) and a right neutral element, they are both equal. {{proofat|[[Neutral element#Any left and right neutral element are equal]]}}
+# A group can be defined as a set with three operations: a binary operation, called the multiplication or product, an inverse map, and an identity element. These three operations are subject to the conditions of associativity, identity element and inverses.
-# Now that we know what the [[trivial group]] is, we can interpret this in a somewhat more sophisticated way. For any group, there is a ''unique'' subgroup of it that looks like the trivial group. That subgroup is the one and only identity element of the group.
+# A group can also be defined as a set with an associative binary operation, called the multiplication or product, such that there exists an identity element and every element has an inverse.
-# In particular, any group is ''nonempty''.
+# The two definitions of group are equivalent because the identity element is unique, and inverses, when they exist, are unique. We'll see proofs of this in part two.
-# Given any element, it has a ''unique'' inverse. The proof of this uses associativity. The proof works in a slightly greater generality: if we have a [[monoid]] (a set with associative binary operation and a [[neutral element]] i.e. a multiplicative identity) and an element ''happens'' to have a left inverse and also ''happens'' to have a right inverse, the left and right inverse must be equal. {{proofat|[[Inverse element#Equality of left and right inverses]]}}
+# Subgroups are subsets that are closed under all the group operations. Just having a subset that is closed under the multiplication operation is ''not'' enough: for instance, the nonzero integers are closed under multiplication inside the group of nonzero rationals under multiplication, but they don't form a subgroup.
-# The upshot: the binary operation of the group determines the other two operations.
+# There are two somewhat different ways of defining subgroups: one, where we demand closure under all the group operations, and the other, where we demand closure only under the multiplication, but insist that the subset form a group with this induced operation. The two definitions are equivalent, and we'll see a proof of this in part two.
-# Subgroups are subsets that are closed under all the group operations. Just having a subset that is closed under the multiplication operation, is ''not'' enough.
+# There is a third way of defining subgroups in terms of left quotients of elements. We'll see, in part two, a proof of the equivalence of this with the other definitions.
-# However, if we have a nonempty subset that is closed under the operation <math>(x,y) \mapsto xy^{-1}</math> it is a subgroup. {{proofat|[[Sufficiency of subgroup condition]]}}
+# A very special class of groups is the [[abelian group]]s. A group is abelian iff any two elements ''commute'' i.e. the binary operation on it is commutative. Any subgroup of an abelian group is abelian.
-# A very special class of groups is the [[Abelian group]]s. A group is Abelian iff any two elements ''commute'' i.e. the binary operation on it is commutative. Any subgroup of an Abelian group is Abelian.
+# The multiplication operation in an abelian group is often denoted additively.
-# The multiplication operation in an Abelian group is denoted additively.
+# The trivial group is the group with just one element.
 ==Notational conventions==
 # Groups are often denoted by letters like <math>G,H,K</math>
-# The multiplication operation is denoted by <math>*</math> or <matH>\cdot</math>, or by omission. Because of associativity of multiplication, we can omit parentheses when multiplying more than two elements.
+# The multiplication operation is denoted by <math>*</math> or <math>\cdot</math>, or by omission. Because of associativity of multiplication, we can omit parentheses when multiplying more than two elements, and, when the context is clear, omit the multiplication symbol as well.
 # The inverse operation is denoted by a superscript of <matH>-1</math>. The superscript applies only to the immediately preceding expression. Thus <math>xy^{-1}</math> is <math>x * (y^{-1})</math> and not <math>(x * y)^{-1}</math>
 # A product of the same element with itself many times is denoted by a power of that element. So <math>x^n = xxx\ldots</math> <math>n</math> times
 # The multiplicative identity is denoted by <math>e</math> or <math>1</math>
-# For Abelian groups, <math>+</math> denotes the addition, and iterated sum is denoted by integer multiplication. So <math>nx = x + x + \ldots + x</math> done <math>n</math> times.
+# For abelian groups, <math>+</math> denotes the addition, and iterated sum is denoted by integer multiplication. So <math>nx = x + x + \ldots + x</math> done <math>n</math> times.
-# <math>-</math> denotes the additive inverse in an Abelian group
+# <math>-</math> denotes the additive inverse in an abelian group, and <math>0</math> denotes the additive identity.
 # Subgroups in general are denoted by the <math>\le</math> sign. So <math>H \le G</math> means <math>H</math> is a subgroup of <math>G</math>. We can also say <math>H \subseteq G</math>, but the latter is also used for mere subsets, that aren't subgroups.
-{{guided tour|beginners|Introduction one|Entertainment menu one (beginners)|Understanding the definition of a group}}
+{{guided tour-bottom|beginners|Introduction one|Entertainment menu one (beginners)|Understanding the definition of a group}}