Editing Group (mathematics) (section)

== Basic concepts ==
{{hatnote|The following sections use [[glossary of mathematical symbols|mathematical symbols]] such as <math>X=\{x,y,z\}</math> to denote a [[set (mathematics)|set]] <math>X</math> containing [[element (mathematics)|elements]] {{tmath|1= x }}, {{tmath|1= y }}, and {{tmath|1= z }}, or <math>x\in X</math> to state that <math>x</math> is an element of {{tmath|1= X }}. The notation <math>f:X\to Y</math> means <math>f</math> is a [[function (mathematics)|function]] associating to every element of <math>X</math> an element of {{tmath|1= Y }}.}}

When studying sets, one uses concepts such as [[subset]], function, and [[quotient by an equivalence relation]]. When studying groups, one uses instead [[subgroup]]s, [[group homomorphism|homomorphism]]s, and [[quotient group]]s. These are the analogues that take the group structure into account.{{efn|See, for example, {{harvnb|Lang|2002}}, {{harvnb|Lang|2005}}, {{harvnb|Herstein|1996}} and {{harvnb|Herstein|1975}}.}}

=== Group homomorphisms ===
{{Main|Group homomorphism}}
Group homomorphisms{{efn|The word homomorphism derives from [[Ancient Greek|Greek]] ὁμός—the same and [[wikt:μορφή|μορφή]]—structure. See {{harvnb|Schwartzman|1994|p=108}}.}} are functions that respect group structure; they may be used to relate two groups. A ''homomorphism'' from a group <math>(G,\cdot)</math> to a group <math>(H,*)</math> is a function <math>\varphi : G\to H</math> such that
{{Block indent|left=1.6|<math>\varphi(a\cdot b)=\varphi(a)*\varphi(b)</math> for all elements <math>a</math> and <math>b</math> in {{tmath|1= G }}.}}
It would be natural to require also that <math>\varphi</math> respect identities, {{tmath|1= \varphi(1_G)=1_H }}, and inverses, <math>\varphi(a^{-1})=\varphi(a)^{-1}</math> for all <math>a</math> in {{tmath|1= G }}. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.{{sfn|Lang|2005|loc=§II.3|p=34}}

The ''identity homomorphism'' of a group <math>G</math> is the homomorphism <math>\iota_G : G\to G</math> that maps each element of <math>G</math> to itself. An ''inverse homomorphism'' of a homomorphism <math>\varphi : G\to H</math> is a homomorphism <math>\psi : H\to G</math> such that <math>\psi\circ\varphi=\iota_G</math> and {{tmath|1= \varphi\circ\psi=\iota_H }}, that is, such that <math>\psi\bigl(\varphi(g)\bigr)=g</math> for all <math>g</math> in <math>G</math> and such that <math>\varphi\bigl(\psi(h)\bigr)=h</math> for all <math>h</math> in {{tmath|1= H }}. An ''[[group isomorphism|isomorphism]]'' is a homomorphism that has an inverse homomorphism; equivalently, it is a [[bijective]] homomorphism. Groups <math>G</math> and <math>H</math> are called ''isomorphic'' if there exists an isomorphism {{tmath|1= \varphi : G\to H }}. In this case, <math>H</math> can be obtained from <math>G</math> simply by renaming its elements according to the function {{tmath|1= \varphi }}; then any statement true for <math>G</math> is true for {{tmath|1= H }}, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a [[category (mathematics)|category]], the [[category of groups]].{{sfn|Mac Lane|1998}}

An [[injective]] homomorphism <math>\phi : G' \to G</math> factors canonically as an isomorphism followed by an inclusion, <math>G' \;\stackrel{\sim}{\to}\; H \hookrightarrow G</math> for some subgroup {{tmath|1= H }} of {{tmath|1= G }}.
Injective homomorphisms are the [[monomorphism]]s in the category of groups.

=== Subgroups ===
{{Main|Subgroup}}
Informally, a ''subgroup'' is a group <math>H</math> contained within a bigger one, {{tmath|1= G }}: it has a subset of the elements of {{tmath|1= G }}, with the same operation.{{sfn|Lang|2005|loc=§II.1|p=19}} Concretely, this means that the identity element of <math>G</math> must be contained in {{tmath|1= H }}, and whenever <math>h_1</math> and <math>h_2</math> are both in {{tmath|1= H }}, then so are <math>h_1\cdot h_2</math> and {{tmath|1= h_1^{-1} }}, so the elements of {{tmath|1= H }}, equipped with the group operation on <math>G</math> restricted to {{tmath|1= H }}, indeed form a group. In this case, the inclusion map <math>H \to G</math> is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup {{tmath|1= R=\{\mathrm{id},r_1,r_2,r_3\} }}, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The [[subgroup test]] provides a [[Necessary and sufficient conditions|necessary and sufficient condition]] for a nonempty subset {{tmath|1= H }} of a group {{tmath|1= G }} to be a subgroup: it is sufficient to check that <math>g^{-1}\cdot h\in H</math> for all elements <math>g</math> and <math>h</math> in {{tmath|1= H }}. Knowing a group's [[lattice of subgroups|subgroups]] is important in understanding the group as a whole.{{efn|However, a group is not determined by its lattice of subgroups. See {{harvnb|Suzuki|1951}}.}}

Given any subset <math>S</math> of a group {{tmath|1= G }}, the subgroup [[Generating set of a group|generated]] by <math>S</math> consists of all products of elements of <math>S</math> and their inverses. It is the smallest subgroup of <math>G</math> containing {{tmath|1= S }}.{{sfn|Ledermann|1973|loc=§II.12|p=39}} In the example of symmetries of a square, the subgroup generated by <math>r_2</math> and <math>f_{\mathrm{v}}</math> consists of these two elements, the identity element {{tmath|1= \mathrm{id} }}, and the element {{tmath|1= f_{\mathrm{h} }=f_{\mathrm{v} }\cdot r_2 }}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

=== Cosets ===
{{Main|Coset}}
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup <math>H</math> determines left and right cosets, which can be thought of as translations of <math>H</math> by an arbitrary group element {{tmath|1= g }}. In symbolic terms, the ''left'' and ''right'' cosets of {{tmath|1= H }}, containing an element {{tmath|1= g }}, are

{{Block indent|left=1.6|<math>gH=\{g\cdot h\mid h\in H\}</math> and {{tmath|1= Hg=\{h\cdot g\mid h\in H\} }}, respectively.{{sfn|Lang|2005|loc=§II.4|p=41}}}}

The left cosets of any subgroup <math>H</math> form a [[Partition of a set|partition]] of {{tmath|1= G }}; that is, the [[Union (set theory)|union]] of all left cosets is equal to <math>G</math> and two left cosets are either equal or have an [[empty set|empty]] [[Intersection (set theory)|intersection]].{{sfn|Lang|2002|loc=§I.2|p=12}} The first case <math>g_1H=g_2H</math> happens [[if and only if|precisely when]] {{tmath|1= g_1^{-1}\cdot g_2\in H }}, i.e., when the two elements differ by an element of {{tmath|1= H }}. Similar considerations apply to the right cosets of {{tmath|1= H }}. The left cosets of <math>H</math> may or may not be the same as its right cosets. If they are (that is, if all <math>g</math> in <math>G</math> satisfy {{tmath|1= gH=Hg }}), then <math>H</math> is said to be a ''[[normal subgroup]]''.

In {{tmath|1= \mathrm{D}_4 }}, the group of symmetries of a square, with its subgroup <math>R</math> of rotations, the left cosets <math>gR</math> are either equal to {{tmath|1= R }}, if <math>g</math> is an element of <math>R</math> itself, or otherwise equal to <math>U=f_{\mathrm{c}}R=\{f_{\mathrm{c}},f_{\mathrm{d}},f_{\mathrm{v}},f_{\mathrm{h}}\}</math> (highlighted in green in the Cayley table of {{tmath|1= \mathrm{D}_4 }}). The subgroup <math>R</math> is normal, because <math>f_{\mathrm{c}}R=U=Rf_{\mathrm{c}}</math> and similarly for the other elements of the group. (In fact, in the case of {{tmath|1= \mathrm{D}_4 }}, the cosets generated by reflections are all equal: {{tmath|1= f_{\mathrm{h} }R=f_{\mathrm{v} }R=f_{\mathrm{d} }R=f_{\mathrm{c} }R }}.)

=== Quotient groups ===
{{Main|Quotient group}}
Suppose that <math>N</math> is a normal subgroup of a group {{tmath|1= G }}, and
<math display=block>G/N = \{gN \mid g\in G\}</math>
denotes its set of cosets.
Then there is a unique group law on <math>G/N</math> for which the map <math>G\to G/N</math> sending each element <math>g</math> to <math>gN</math> is a homomorphism.
Explicitly, the product of two cosets <math>gN</math> and <math>hN</math> is {{tmath|1= (gh)N }}, the coset <math>eN = N</math> serves as the identity of {{tmath|1= G/N }}, and the inverse of <math>gN</math> in the quotient group is {{tmath|1= (gN)^{-1} = \left(g^{-1}\right)N }}.
The group {{tmath|1= G/N }}, read as "{{tmath|1= G }} modulo {{tmath|1= N }}",{{sfn|Lang|2005|loc=§II.4|p=45}} is called a ''quotient group'' or ''factor group''.
The quotient group can alternatively be characterized by a [[universal property]].

{| class="wikitable" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:200px;"
|+ Cayley table of the quotient group <math>\mathrm{D}_4/R</math>
|-
! style="width:30px;"| <math>\cdot</math>
! style="width:33%;"| <math>R</math>
! style="width:33%;"| <math>U</math>
|-
! <math>R</math>
| <math>R</math> || <math>U</math>
|-
! <math>U</math>
| <math>U</math> || <math>R</math>
|}
The elements of the quotient group <math>\mathrm{D}_4/R</math> are <math>R</math> and {{tmath|1= U=f_{\mathrm{v} }R }}. The group operation on the quotient is shown in the table. For example, {{tmath|1= U\cdot U=f_{\mathrm{v} }R\cdot f_{\mathrm{v} }R=(f_{\mathrm{v} }\cdot f_{\mathrm{v} })R=R }}. Both the subgroup <math>R=\{\mathrm{id},r_1,r_2,r_3\}</math> and the quotient <math>\mathrm{D}_4/R</math> are abelian, but <math>\mathrm{D}_4</math> is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the [[semidirect product]] construction; <math>\mathrm{D}_4</math> is an example.

The [[first isomorphism theorem]] implies that any [[surjective]] homomorphism <math>\phi : G \to H</math> factors canonically as a quotient homomorphism followed by an isomorphism: {{tmath|1= G \to G/\ker \phi \;\stackrel{\sim}{\to}\; H }}.
Surjective homomorphisms are the [[epimorphism]]s in the category of groups.

=== Presentations ===
{{Main|Presentation of a group}}
Every group is isomorphic to a quotient of a [[free group]], in many ways.

For example, the dihedral group <math>\mathrm{D}_4</math> is generated by the right rotation <math>r_1</math> and the reflection <math>f_{\mathrm{v}}</math> in a vertical line (every element of <math>\mathrm{D}_4</math> is a finite product of copies of these and their inverses).
Hence there is a surjective homomorphism {{tmath|1= \phi }} from the free group <math>\langle r,f \rangle</math> on two generators to <math>\mathrm{D}_4</math> sending <math>r</math> to <math>r_1</math> and <math>f</math> to {{tmath|1= f_1 }}.
Elements in <math>\ker \phi</math> are called ''relations''; examples include {{tmath|1= r^4,f^2,(r \cdot f)^2 }}.
In fact, it turns out that <math>\ker \phi</math> is the smallest normal subgroup of <math>\langle r,f \rangle</math> containing these three elements; in other words, all relations are consequences of these three.
The quotient of the free group by this normal subgroup is denoted {{tmath|1= \langle r,f \mid r^4=f^2=(r\cdot f)^2=1 \rangle }}.
This is called a ''[[presentation of a group|presentation]]'' of <math>\mathrm{D}_4</math> by generators and relations, because the first isomorphism theorem for {{tmath|1= \phi }} yields an isomorphism {{tmath|1= \langle r,f \mid r^4=f^2=(r\cdot f)^2=1 \rangle \to \mathrm{D}_4 }}.{{sfn|Lang|2002|loc=§I.2|p=9}} 

A presentation of a group can be used to construct the [[Cayley graph]], a graphical depiction of a [[discrete group]].{{sfn|Magnus|Karrass|Solitar|2004|loc=§1.6|pp=56–67}}