Difference between revisions of "Tour:Factsheet one (beginners)"

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{{guided tour|beginners|Introduction one|Entertainment menu one (beginners)|Understanding the definition of a group}}
 
{{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.
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Here are some facts we have seen so far:
  
# The identity element in a group is unique. We'll see the proof of this in part two.
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# 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.
# The trivial group is the group with just one element. In every group, we can find a copy of the trivial group -- as the subset with just the identity element.
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# A group can also be defiend as a set with one 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'', and the smallest group is the trivial group.
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# 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. We'll see this proof in part two.
 
# The binary operation of a group determines the other two operations.
 
 
# 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.
 
# 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.
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# 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.
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# 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 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 often denoted additively.
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# The trivial group is the group with just one element.
  
 
==Notational conventions==
 
==Notational conventions==

Revision as of 16:39, 26 July 2008

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Understanding the definition of a group |UP: Introduction one (beginners) | NEXT: Entertainment menu one (beginners)

Here are some facts we have seen so far:

  1. 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.
  2. A group can also be defiend as a set with one binary operation, called the multiplication or product, such that there exists an identity element and every element has an inverse.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. The multiplication operation in an Abelian group is often denoted additively.
  9. The trivial group is the group with just one element.

Notational conventions

  1. Groups are often denoted by letters like G,H,K
  2. 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.
  3. The inverse operation is denoted by a superscript of -1. The superscript applies only to the immediately preceding expression. Thus xy^{-1} is x * (y^{-1}) and not (x * y)^{-1}
  4. A product of the same element with itself many times is denoted by a power of that element. So x^n = xxx\ldots n times
  5. The multiplicative identity is denoted by e or 1
  6. For Abelian groups, + denotes the addition, and iterated sum is denoted by integer multiplication. So nx = x + x + \ldots + x done n times.
  7. - denotes the additive inverse in an Abelian group, and 0 denotes the additive identity.
  8. Subgroups in general are denoted by the \le sign. So H \le 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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