Elementary group theory

In mathematics, a group <"G",*> is defined as a set "G" and a binary operation on "G", called "product" and denoted by infix "*". Product obeys the following rules (also called axioms). Let "a", "b", and "c" be arbitrary elements of "G". Then:
*A1, Closure. "a"*"b" is in "G";
*A2, Associativity. ("a"*"b")*"c"="a"*("b"*"c");
*A3, Identity. There exists an identity element "e" in "G" such that "a"*"e"="e"*"a"="a". "e", the "identity" of "G", is unique by Theorem 1.4 below;
*A4, Inverse. For each "a" in "G", there exists an inverse element "x" in "G" such that "a"*"x"="x"*"a"="e". "x", the "inverse" of "a", is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:
*A5, Commutativity. "a"*"b" = "b"*"a".

Closure is part of the definition of "binary operation," so that A1 is often omitted.

Elaboration

*Group product "*" is not necessarily multiplication. Addition works just as well, as do many less standard operations.
*When * is a standard operation, we use the standard symbol instead (for example, + for addition).
*When * is addition or any commutative operation (except multiplication), 0 usually denotes the identity and -"a" denotes the inverse of "a". The operation is always denoted by something other than * -- often + -- to avoid confusion with multiplication.
*When * is multiplication or a noncommutative operation,"a"*"b" is often written "ab". 1 usually denotes the identity element, and "a" -1 usually denotes the inverse of "a".
*The group <"G",*> is often referred to as "the group "G" or simply "G"; but the operation "*" is fundamental to the description of the group.
*<"G",*> is usually pronounced "the group "G" under *". When affirming that "G" is a group (for example, in a theorem), we say that "G" is a group under *".

Examples

"G"={1,-1} is a group under multiplication, because for all elements "a", "b", "c" in "G"::A1: "a"*"b" is an element of "G".:A2: ("a"*"b")*"c"="a"*("b"*"c") can be verified by enumerating all 8 possible (and trivial) cases.:A3: "a"*1="a". Hence 1 is an identity element.:A4: "a"-1*"a"=1. Hence "a"-1 denotes inverse and 1 is an inverse element.

The integers Z and the real numbers R are groups under addition '+'. For all elements "a", "b", and "c" of either Z or R::A1: Adding any two numbers yields another number of the same kind.:A2: ("a"+"b")+"c"="a"+("b"+"c").:A3: "a"+0="a". Hence 0 is an identity element.:A4: -"a"+"a"=0. Hence -"a" denotes inverse and 0 is an inverse element.

The real numbers R are NOT a group under multiplication '*'. For all "a", "b", and "c" in R::A3: 1.:A4: 0*"a"=0, so 0 has no inverse.The real numbers without 0, R#, are a group under multiplication '*'.:A1: Multiplying any two elements of R# yields another element of R#.:A2: ("a"*"b")*"c"="a"*("b"*"c").:A3: "a"*1="a". Hence 1 denotes identity.:A4: "a" -1*"a"=1. Hence "a" -1 denotes inverse.

Alternative Axioms

A3 and A4 can be replaced by:
*A3&rsquo;, left neutral. There exists an "e" in "G" such that for all "a" in "G", "e"*"a"="a".
*A4&rsquo;, left inverse. For each "a" in "G", there exists an element "x" in "G" such that "x"*"a"="e".

Or alternatively by:
*A3&rsquo;&rsquo;, right neutral. There exists an "e" in "G" such that for all "a" in "G", "a"*"e"="a".
*A4&rsquo;&rsquo;, right inverse. For each "a" in "G", there exists an element "x" in "G" such that "a"*"x"="e".

These apparently weaker pairs of axioms are naturally implied by A3 and A4. We will now show that the converse is true.

Theorem: Assuming A1 and A2, A3&rsquo; and A4&rsquo; imply A3 and A4.

"Proof". Let a left neutral element "e" be given, and "a" in "G". By A4&rsquo; there exist an "x" such that "x"*"a"="e".

We show that also "a"*"x"="e".Per A4&rsquo; there is an "y" in "G" with:: y * (a * x) = e quad (1)

Therefore::egin{align}e & = y * (a * x) &quad (1) \ & = y * (a * (e * x)) &quad (A3') \ & = y * (a * ((x * a) * x)) &quad (A4') \ & = y * (a * (x * (a * x))) &quad (A2) \ & = y * ((a * x) * (a * x)) &quad (A2) \ & = (y * (a * x)) * (a * x) &quad (A2) \ & = e * (a * x) &quad (1) \ & = a * x &quad (A3') \end{align}This establishes A4.

:egin{align}a * e & = & a * (x * a) &quad (A4) \ & = & (a * x) * a &quad (A2) \ & = & e * a &quad (A4) \end{align}This establishes A3.

Theorem: Assuming A1 and A2, A3&rsquo;&rsquo; and A4&rsquo;&rsquo; imply A3 and A4.

"Proof". Similar to the above.

Basic theorems

Identity is unique

Theorem 1.4: The identity element of a group <"G",*> is unique.

"Proof": Suppose that "e" and "f" are two identity elements of "G". Then:egin{align}e & = & e * f &quad (A3") \ & = & f &quad (A3') \end{align}

As a result, we can speak of "the" identity element of <"G",*> rather than "an" identity element. Where different groups are being discussed and compared, "e""G" denotes the identity of the specific group <"G",*>.

Inverses are unique

Theorem 1.5: The inverse of each element in <"G",*> is unique.

"Proof": Suppose that "h" and "k" are two inverses of an element "g" of "G". Then:egin{align}h & = & h * e &quad (A3) \ & = & h * (g * k) &quad (A4) \ & = & (h * g) * k &quad (A2) \ & = & (e * k) &quad (A4) \ & = & k &quad (A3) \end{align}

As a result, we can speak of "the" inverse of an element "a", rather than "an" inverse. Without ambiguity, for all "a" in "G", we denote by "a" -1 the unique inverse of "a".

Latin square property

Theorem 1.3: For all elements "a","b" in "G", there exists a unique "x" in "G" such that "a"*"x" = "b".

"Proof". At least one such "x" surely exists, for if we let "x" = "a" -1*"b", then "x" is in "G" (by A1, closure) and:
* "a"*"x" = "a"*("a" -1*"b") (substituting for "x")
* "a"*("a" -1*"b") = ("a"*"a" -1)*"b" (associativity A2).
*("a"*"a" -1)*"b"= "e"*"b" = "b". (identity A3).
* Thus an "x" always exists satisfying "a"*"x" = "b".To show that this is unique, if "a"*"x"="b", then
* "x" = "e"*"x"
* "e"*"x" = ("a" -1*"a")*x
* ("a" -1*"a")*x = "a" -1*("a"*"x")
* "a" -1*("a"*"x") = "a" -1*"b"
* Thus, "x" = "a" -1*"b"Similarly, for all "a","b" in "G", there exists a unique "y" in "G" such that "y"*"a" = "b".

Inverting twice gets you back where you started

Theorem 1.6: For all elements "a" in group "G", ("a" -1) -1="a".

"Proof". "a" -1*"a" = "e". The conclusion follows from Theorem 1.4.

Inverse of "ab"

Theorem 1.7: For all elements "a","b" in group "G", ("a"*"b") -1="b" -1*"a" -1.

"Proof". ("a"*"b")*("b" -1*"a" -1) = "a"*("b"*"b" -1)*"a" -1 = "a"*e*"a" -1 = "a"*"a" -1 = "e". The conclusion follows from Theorem 1.4.

Cancellation

Theorem 1.8: For all elements "a","x", and "y" in group "G", if "a"*"x"="a"*"y", then "x"="y"; and if "x"*"a"="y"*"a", then "x"="y".

"Proof". If "a"*"x" = "a"*"y" then:
* "a" -1*("a"*"x") = "a" -1*("a"*"y")
* ("a" -1*"a")*"x" = ("a" -1*"a")*"y"
* "e"*"x" = "e"*"y"
* "x" = "y"If "x"*"a" = "y"*"a" then
* ("x"*"a")*"a" -1 = ("y"*"a")*"a" -1
* "x"*("a"*"a" -1) = "y"*("a"*"a" -1)
* "x"*"e" = "y"*"e"
* "x" = "y"

Powers

For n in mathbb{Z} and a in G we define::a ^ n :=egin{cases}underbrace{a*a*cdots*a}_{n mbox{times, & mbox{if }n > 0 \1, & mbox{if }n = 0 \underbrace{a^{-1}*a^{-1}*cdots*a^{-1_{-n mbox{times, & mbox{if }n < 0end{cases}

Theorem 1.9: For all "a" in group <"G",*>, n, m in mathbb{Z}::egin{matrix}a^m*a^n &=& a^{m+n}\(a^m)^n &=& a^{m*n}end{matrix}

Similarly if "G" is written in additive notation, we have::n * a :=egin{cases}underbrace{a+a+cdots+a}_{n mbox{times, & mbox{if }n > 0 \0, & mbox{if }n = 0 \underbrace{(-a)+(-a)+cdots+(-a)}_{-n mbox{times, & mbox{if }n < 0end{cases}

and::egin{matrix}(m*a) +(n*a) &=& (m+n)*a\m*(n*a) &=& (m*n)*aend{matrix}

Order

Of a group element

The order of an element "a" in a group "G" is the least positive integer "n" such that "an = e". Sometimes this is written "o("a")="n". "n" can be infinite.

Theorem 1.10: A group whose nontrivial elements all have order 2 is abelian. In other words, if all elements "g" in a group "G" "g"*"g"="e" is the case, then for all elements "a","b" in "G", "a"*"b"="b"*"a".

"Proof". Let "a", "h" be any 2 elements in the group "G". By A1, "a"*"h" is also a member of "G". Using the given condition, we know that ("a"*"h")*("a"*"h")="e". Hence:
* "a"*("a"*"b")*("a"*"b") = "a"*"e"
* "a"*("a"*"b")*("a"*"b")*"b" = "a"*"e"*"b"
* ("a"*"a")*("b"*"a")*("b"*"b") = ("a"*"e")*"b"
* "e"*("b"*"a")*"e" = "a"*"b"
* "b"*"a" = "a"*"b".Since the group operation * commutes, the group is abelian

Of a group

The order of the group "G", usually denoted by |"G"| or occasionally by o("G"), is the number of elements in the set "G", in which case <"G",*> is a "finite group". If "G" is an infinite set, then the group <"G",*> has order equal to the cardinality of "G", and is an "infinite group".

ubgroups

A subset "H" of "G" is called a subgroup of a group <"G",*> if "H" satisfies the axioms of a group, using the same operator "*", and restricted to the subset "H". Thus if "H" is a subgroup of <"G",*>, then <"H",*> is also a group, and obeys the above theorems, restricted to "H". The "order" of subgroup "H" is the number of elements in "H".

A "proper subgroup" of a group "G" is a subgroup which is not identical to "G". A "non-trivial" subgroup of "G" is (usually) any proper subgroup of "G" which contains an element other than "e".

Theorem 2.1: If "H" is a subgroup of <"G",*>, then the identity "e""H" in "H" is identical to the identity "e" in ("G",*).

"Proof". If "h" is in "H", then "h"*"e""H" = "h"; since "h" must also be in "G", "h"*"e" = "h"; so by theorem 1.4, "e""H" = "e".

Theorem 2.2: If "H" is a subgroup of "G", and "h" is an element of "H", then the inverse of "h" in "H" is identical to the inverse of "h" in "G".

"Proof". Let "h" and "k" be elements of "H", such that "h"*"k" = "e"; since "h" must also be in "G", "h"*"h" -1 = "e"; so by theorem 1.5, "k" = "h" -1.

Given a subset "S" of "G", we often want to determine whether or not "S" is also a subgroup of "G". A handy theorem valid for both infinite and finite groups is:

Theorem 2.3: If "S" is a non-empty subset of "G", then "S" is a subgroup of "G" if and only if for all "a","b" in "S", "a"*"b" -1 is in "S".

"Proof". If for all "a", "b" in "S", "a"*"b" -1 is in "S", then
* "e" is in "S", since "a"*"a" -1 = "e" is in "S".
* for all "a" in "S", "e"*"a" -1 = "a" -1 is in "S"
* for all "a", "b" in "S", "a"*"b" = "a"*("b" -1) -1 is in "S"Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so "S" is subgroup.

Conversely, if "S" is a subgroup of "G", then it obeys the axioms of a group.
* As noted above, the identity in "S" is identical to the identity "e" in "G".
* By A4, for all "b" in "S", "b" -1 is in "S"
* By A1, "a"*"b" -1 is in "S".

The intersection of two or more subgroups is again a subgroup.

Theorem 2.4: The intersection of any non-empty set of subgroups of a group "G" is a subgroup.

"Proof". Let {"H""i"} be a set of subgroups of "G", and let K = ∩{"H""i"}. "e" is a member of every "H""i" by theorem 2.1; so "K" is not empty. If "h" and "k" are elements of "K", then for all "i",
* "h" and "k" are in "H""i".
* By the previous theorem, "h"*"k" -1 is in "H""i"
* Therefore, "h"*"k" -1 is in ∩{"H""i"}.Therefore for all "h", "k" in "K", "h"*"k" -1 is in "K". Then by the previous theorem, "K"=∩{"H""i"} is a subgroup of "G"; and in fact "K" is a subgroup of each "H""i".

Given a group <"G",*>, define "x"*"x" as "x"², "x"*"x"*"x"*...*"x" ("n" times) as "x""n", and define "x"0 = "e". Similarly, let "x" -"n" for ("x" -1)"n". Then we have:

Theorem 2.5: Let "a" be an element of a group ("G",*). Then the set {"a""n": "n" is an integer} is a subgroup of "G".

"Proof". A subgroup of this type is called a "cyclic" subgroup; the subgroup of the powers of "a" is often written as <"a">, and we say that "a" "generates" <"a">.

Cosets

If "S" and "T" are subsets of "G", and "a" is an element of "G", we write "a"*"S" to refer to the subset of "G" made up of all elements of the form "a"*"s", where "s" is an element of "S"; similarly, we write "S"*"a" to indicate the set of elements of the form "s"*"a". We write "S"*"T" for the subset of "G" made up of elements of the form "s"*"t", where "s" is an element of "S" and "t" is an element of "T".

If "H" is a subgroup of "G", then a "left coset" of "H" is a set of the form "a"*"H", for some "a" in "G". A "right coset" is a subset of the form "H"*"a".

If "H" is a subgroup of "G", the following useful theorems, stated without proof, hold for all cosets:

* And "x" and "y" are elements of "G", then either "x"*"H" = "y"*"H", or "x"*"H" and "y"*"H" have empty intersection.

* Every left (right) coset of "H" in "G" contains the same number of elements.

* "G" is the disjoint union of the left (right) cosets of "H".

* Then the number of distinct left cosets of "H" equals the number of distinct right cosets of "H".

Define the index of a subgroup "H" of a group "G" (written " ["G":"H"] ") to be the number of distinct left cosets of "H" in "G".

From these theorems, we can deduce the important Lagrange's theorem, relating the order of a subgroup to the order of a group:

* Lagrange's theorem: If "H" is a subgroup of "G", then |"G"| = |"H"|* ["G":"H"] .

For finite groups, this can be restated as:

*Lagrange's theorem: If "H" is a subgroup of a finite group "G", then the order of "H" divides the order of "G".

*If the order of group "G" is a prime number, "G" is cyclic.

ee also

*group theory
*abelian groups
*Glossary of group theory
*List of group theory topics

References

* Jordan, C. R and D.A. "Groups". Newnes (Elsevier), ISBN 0-340-61045-X
* Scott, W R. "Group Theory". Dover Publications, ISBN 0-486-65377-3


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