## Quotient groups

Let $N$ be a normal subgroup of $G$.

• The quotient group is the pair $\left(G/N,\pi \right)$ where
$G/N$ is a group       and       $\pi :G\to G/N$ is a group homomorphism
and the pair $\left(G/N,\pi \right)$ is such that
1. If $K$ is a group and $f:G\to K$ is a homomorphism such that $\mathrm{ker}f\supseteq N$
2. then there is a unique morphism $\stackrel{\sim }{f}:G/N\to K$ such that
commutes.

#### Construction:

A left coset of $N$ in $G$ is a set $gN={gn | n∈N}, whereg∈G .$ Define a group with set $G/N={cosets gN |g∈G}$ and operation $G/N×G/N\to G/N$ given by $(g1N)(g2 N) =g1g2N,$ and let $\pi :G\to G/N$ be the group homomorphism given by $π(g)=gN.$

## Quotient rings

Let $R$ be a ring and let $I$ be an ideal of $R$.

• The quotient is the pair $\left(R/I,\pi \right)$ where
$R/I$ is a ring       and       $\pi :R\to R/I$ is a ring homomorphism
and the pair $\left(R/I,\pi \right)$ is such that
1. If $S$ is a ring and $f:R\to S$ is a ring homomorphism such that $\mathrm{ker}f\supseteq I$
2. then there is a unique morphism $\stackrel{\sim }{f}:R/I\to S$ such that
commutes.

#### Construction:

A coset of $I$ in $R$ is a set $r+I={r+i | i∈I}, wherer∈R .$ Define a ring with set $R/I={cosets r+I |r∈R}$ and operations $+:R/I×R/I\to R/I$ and $\cdot :R/I×R/I\to R/I$ given by $(r1+I) +(r2+I) =r1+ r2+I and (r1+I) (r2+I) =r1r2+I,$ and let $\pi :R\to R/I$ be the ring homomorphism given by $π(r)=r+I.$

$\left(R/I,\pi \right)$ is the quotient of $R$ by $I$.

## Quotient modules

Let $M$ be an $R-$module and let $N$ be an $R-$submodule of $M$.

• The quotient is the pair $\left(M/N,\pi \right)$ where
$M/N$ is an $R-$module       and       $\pi :M\to M/N$ is an $R-$module homomorphism
and the pair $\left(M/N,\pi \right)$ is such that
1. If $P$ is an $R-$module and $f:M\to P$ is an $R-$module homomorphism such that $\mathrm{ker}f\supseteq N$
2. then there is a unique morphism $\stackrel{\sim }{f}:M/N\to P$ such that
commutes.

#### Construction:

A coset of $N$ in $M$ is a set $m+N={m+n | n∈N}, wherem∈M .$ Define an $R-$module with set $M/N={cosets m+N |m∈M}$ and operations $+:M/N×M/N\to M/N$ and $\cdot :R×M/N\to M/N$ given by $(m1+N) +(m2+N) =m1+ m2+N and r⋅(m+N) =r⋅m+N.$ Let $\pi :M\to M/N$ be the $R-$module homomorphism given by $π(m)=m+N.$

$\left(M/N,\pi \right)$ is the quotient of $M$ by $N$.

## Ring of fractions

Let $R$ be a commutative ring and $S$ a subset of $R$.

• The ring of fractions of $R$ with denominators in $S$ is the pair $\left(Q,\iota \right)$, where
$Q$ is a ring       and       $\iota :R\to Q$ is a ring homomorphism
and the pair $\left(Q,\iota \right)$ is such that:
1. If $F$ is a ring and $f:R\to F$ is a ring homomorphism such that all elements of $f\left(S\right)$ are invertible in $F$
2. then there is a unique morphism (of rings) $\stackrel{˜}{f}:Q\to F$ such that
commutes.

#### Construction:

Define a ring with set $Q=\left\{\phantom{\rule{.5em}{0ex}}\frac{a}{b}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}a\in R,\phantom{\rule{.5em}{0ex}}b\in S\text{'}\right\}/=$ where $S\text{'}$ is the multiplicatively closed set containing and generated by $S$, $=$ is the equivalence relation given by

$\frac{a}{b}=\frac{c}{d}$       if       $ads=bcs,$ for some $s\in S\text{'}$,
and with operations given by
$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$       and       $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.$
Let $\iota :R\to Q$ be the ring homomorphism given by $ι(r)= r 1 .$

$Q$ is the ring of fractions of $R$ with denominators in $S$.

## Group products

Let $\left({G}_{i}\right)_{i}{}_{\in }{}_{I}$ be a family of groups. The product $\prod _{i\in I}{G}_{i}$ is the pair $∏ i∈I Gi,(πi)i∈I$ where

• $\prod _{i\in I}{G}_{i}$ is a group, and
• $\left({\pi }_{i}:\prod _{i\in I}{G}_{i}\to {G}_{i}\right)$ is a family of homomorphisms
such that $\left({\prod }_{i\in I}{G}_{i},\left({\pi }_{i}{\right)}_{i\in I}\right)$ satisfies
1. if $K$ is a group and ${f}_{i}:K\to {G}_{i}$ is a family of homomorphisms
2. then there is a unique homomorphism $\stackrel{˜}{f}:K\to {\prod }_{i\in I}{G}_{i}$ such that
commutes for all $i$.

#### Construction:

Define a group with set $∏ i∈I Gi = {(gi)i∈I | gi∈Gi}$ and operation $⋅: ∏i∈IGi× ∏i∈IGi →∏i∈IGi$ given by $(gi)i∈I⋅(hi=(gihi).$ Define maps ${\pi }_{i}:\prod _{i\in I}{G}_{i}\to {G}_{i}$ by $πi (gj) =gi.$

$\left(\prod _{i\in I}{G}_{i},\left({\pi }_{i}\right)\right)$ is the product group for the family of groups $\left({G}_{i}{\right)}_{i\in I}$.

## Restricted products

Let $\left({G}_{i}{\right)}_{i\in I}$ be a family of groups. The (restricted) product is the pair $∏ i∈I Gi (ιi)$ where

• $\prod _{i\in I}{G}_{i}$ is a group, and
• ${\iota }_{i}:{G}_{i}\to \prod _{i\in I}{G}_{i}$ is a family of homomorphisms
such that
1. if $K$ is a group and ${f}_{i}:{G}_{i}\to K$ is a family of homomorphisms such that ${f}_{i}\left(g\right)$ and ${f}_{j}\left(h\right)$ commute for all $i\ne j$, $g\in {G}_{i}$, $h\in {G}_{j}$
2. then there is a unique morphism $\stackrel{˜}{f}:\prod _{i\in I}{G}_{i}\to K$ such that
commutes for all $i$.

#### Construction:

Define a group with set $∏ i∈I Gi = (gi)i∈I gi∈Gi and all but finitely many of the gi are equal to 1$ and operation $\cdot :\prod _{i\in I}{G}_{i}×\prod _{i\in I}{G}_{i}\to \prod _{i\in I}{G}_{i}$ given by $(gi) (gi') = (gigi').$ Define ${\iota }_{i}:{G}_{i}\to \prod _{i\in I}{G}_{i}$ by

$\left(\prod _{i\in I}{G}_{i},{\iota }_{i}\right)$ is the (restricted) product of the family of groups $\left({G}_{i}{\right)}_{i\in I}$.

## Inverse limits

Let $I$ be a set with a preordering $\alpha \le \beta$. Let $\left({E}_{\alpha },\pi _{\alpha }{}_{\beta }\right)$ be an inverse system of groups. The inverse limit is the pair $lim ← Eα ( πα ) α∈I$ where

• $\underset{←}{\mathrm{lim}}{E}_{\alpha }$ is a group, and
• ${\pi }_{\alpha }:\underset{←}{\mathrm{lim}}{E}_{\alpha }\to {E}_{\alpha }$ are morphisms such that ${\pi }_{\alpha }=\pi _{\alpha }{}_{\beta }\circ {\pi }_{\beta }$
and the pair $\left(\underset{←}{\mathrm{lim}}{E}_{\alpha },\left({\pi }_{\alpha }{\right)}_{\alpha \in I}\right)$ is such that
1. if $G$ is a group and ${f}_{\alpha }:G\to {E}_{\alpha }$ are morphisms such that ${f}_{\alpha }=\pi _{\alpha }{}_{\beta }\circ {f}_{\beta }$
2. then there is a unique morphism $\stackrel{˜}{f}:G\to \underset{←}{\mathrm{lim}}{E}_{\alpha }$ such that
commutes for all $\alpha$.

#### Construction:

Define a group with set $lim ← Eα = (gα) π α β ( gβ) = gα for α≤β$ and operation $\cdot :\underset{←}{\mathrm{lim}}{E}_{\alpha }×\underset{←}{\mathrm{lim}}{E}_{\alpha }\to \underset{←}{\mathrm{lim}}{E}_{\alpha }$ given by $(gα)⋅ (gα') = ( gα gα').$ Define ${\pi }_{\alpha }:\underset{←}{\mathrm{lim}}{E}_{\alpha }\to {E}_{\alpha }$ by $πα (( gβ )) = gα.$

$\left(\underset{←}{\mathrm{lim}}{E}_{\alpha },\left({\pi }_{\alpha }{\right)}_{\alpha \in I}\right)$ is the inverse limit of the inverse system $\left(\left({E}_{\alpha }{\right)}_{\alpha \in I},\pi _{\alpha }{}_{\beta }\right).$

## Direct limits

Let $\left({E}_{\alpha },\iota _{\beta }{}_{\alpha }\right)$ be a direct system of groups. The direct limit is the pair $lim → Eα ια$ where

• $\underset{\to }{\mathrm{lim}}{E}_{\alpha }$ is a group, and
• ${\iota }_{\alpha }:{E}_{\alpha }\to \underset{\to }{\mathrm{lim}}{E}_{\alpha }$ are morphisms such that ${\iota }_{\beta }\circ \iota _{\beta }{}_{\alpha }={\iota }_{\alpha }$
and $\left(\underset{\to }{\mathrm{lim}}{E}_{\alpha },{\iota }_{\alpha }\right)$ is such that
1. if $G$ is a group and ${f}_{\alpha }:{E}_{\alpha }\to G$ are morphisms such that ${f}_{\beta }\circ \iota _{\beta }{}_{\alpha }={f}_{\alpha }$ for all $\alpha$
2. then there is a unique morphism $\stackrel{˜}{f}:\underset{\to }{\mathrm{lim}}{E}_{\alpha }\to G$ such that
commutes for all $\alpha$.

#### Construction:

Define a group with set $lim → Eα = (gα) gα∈ Eα, and all but finitely many gα are 1 /~$ where $~$ is the relation given by and operation $\cdot :\underset{\to }{\mathrm{lim}}{E}_{\alpha }×\underset{\to }{\mathrm{lim}}{E}_{\alpha }\to \underset{\to }{\mathrm{lim}}{E}_{\alpha }$ given by $(gα)⋅ (gβ)= (gα gβ).$ Define ${\iota }_{\alpha }:{E}_{\alpha }\to \underset{\to }{\mathrm{lim}}{E}_{\alpha }$ by $ια (g)= 1 ... 1 g 1 ... .$

$\left(\underset{\to }{\mathrm{lim}}{E}_{\alpha },{\iota }_{\alpha }\right)$ is the direct limit of the direct system $\left({E}_{\alpha },{\iota }_{\alpha }\right)$.

## Fiber products

Let ${\phi }_{1}:{G}_{1}\to H$ and ${\phi }_{2}:{G}_{2}\to H$ be group homomorphisms. The fiber product is the triplet $G1×H G2 π1 π2$ where

• ${G}_{1}{×}_{H}{G}_{2}$ is a group, and
• ${\pi }_{1}:{G}_{1}{×}_{H}{G}_{2}\to {G}_{1}$ and ${\pi }_{2}:{G}_{1}{×}_{H}{G}_{2}\to {G}_{2}$ are homomorphisms such that ${\phi }_{1}\circ {\pi }_{1}={\phi }_{2}\circ {\pi }_{2}$
and the triplet $\left({G}_{1}{×}_{H}{G}_{2},{\pi }_{1},{\pi }_{2}\right)$ is such that
1. if $K$ is a group and ${f}_{1}:K\to {G}_{1}$ and ${f}_{2}:K\to {G}_{2}$ are homomorphisms such that ${\phi }_{1}\circ {f}_{1}={\phi }_{2}\circ {f}_{2}$
2. then there is a unique morphism $\stackrel{˜}{f}:K\to {G}_{1}{×}_{H}{G}_{2}$ such that
commutes for each $i$.

#### Construction:

Define a group with set $G1 ×H G2 = g1 g2 ∈ G1× G2 φ1 (g1)= φ2 (g2)$ and operation $\cdot :\left({G}_{1}{×}_{H}{G}_{2}\right)×\left({G}_{1}{×}_{H}{G}_{2}\right)\to \left({G}_{1}{×}_{H}{G}_{2}\right)$ given by $g1 g2 ⋅ h1 h2 = g1 h1 g2 h2 .$ Let ${\pi }_{1}:{G}_{1}{×}_{H}{G}_{2}\to {G}_{1}$ and ${\pi }_{2}:{G}_{1}{×}_{H}{G}_{2}\to {G}_{2}$ be given by

$\left({G}_{1}{×}_{H}{G}_{2},{\pi }_{1},{\pi }_{2}\right)$ is the fiber product.

## Semidirect products

An extension of $G$ by $F$ is an exact sequence $1→F →ιE →πG →1.$

A central extension of $G$ by $F$ is an extension of $G$ by $F$ such that $ι(F) ⊆Z(E).$

Let $F$ and $G$ be groups, and $\tau :G\to \mathrm{Aut}\left(F\right)$ a morphism. The semidirect product is the pair $\left(F{×}_{\tau }G,s\right)$ where

• $1\to F\stackrel{\iota }{\to }F{×}_{\tau }G\stackrel{\pi }{\to }G\to 1$ is an extension, and
• $s:G\to F{×}_{\tau }G$ is a section of $\pi$
such that $\left(F{×}_{\tau }G,s\right)$ satisfies
1. if $1\to F\stackrel{i}{\to }K\stackrel{p}{\to }G\to 1$ is an extension and $s\text{'}:G\to K$ is a section of $p$
2. then there is a unique morphism $\stackrel{˜}{f}:F{×}_{\tau }G\to K$ such that
commutes, and
commutes.

#### Construction:

Define a group with set $F×τG = g k ∈ F×G =F×G$ and operation $\cdot :\left(F{×}_{\tau }G\right)×\left(F{×}_{\tau }G\right)\to F{×}_{\tau }G$ given by Define $ι: F→ F×τG by ι(g) = g1 , π: F×τG →G by π( gk )=k, and s:G→ F×τG by s(k)= 1k .$

$\left(F{×}_{\tau }G,s\right)$ is the semidirect product of $F$ with $G$ with respect to $\tau$.

## Amalgamated products

Let $\left({G}_{i}{\right)}_{i\in I}$ be a family of groups with $A$ as a subgroup. Let ${h}_{i}:A\to {G}_{i}$ be the injections of $A$ into ${G}_{i}$. The amalgamated product is the pair $*A Gi ιi$ where

• ${*}_{A}{G}_{i}$ is a group, and
• ${\iota }_{i}:{G}_{i}\to {*}_{A}{G}_{i}$ are morphisms such that ${\iota }_{i}{|}_{A}={\iota }_{j}{|}_{A}$ for all $i\ne j$
and the pair $\left({*}_{A}{G}_{i},{\iota }_{i}\right)$ satisfies
1. if $K$ is a group and ${f}_{i}:{G}_{i}\to K$ are morphisms such that ${f}_{i}{|}_{A}={f}_{j}{|}_{A}$ for all $i\ne j$
2. then there is a unique morphism $\stackrel{˜}{f}:{*}_{A}{G}_{i}\to K$ such that
commutes for all $i\in I$.

#### Construction:

Define a group with set $*A Gi = ( gi ) gi ∈ Gi /~$ where $~$ is the relation given by and with operation $\cdot :\left({*}_{A}{G}_{i}\right)×\left({*}_{A}{G}_{i}\right)\to {*}_{A}{G}_{i}$ given by $( gi ) ( gi' ) = ( gi gi' ) .$ Define ${\iota }_{i}:{G}_{i}\to {*}_{A}{G}_{i}$ by

$\left({*}_{A}{G}_{i},\left({\iota }_{i}\right)\right)$ is the amalgamated product for the family of groups $\left({G}_{i}{\right)}_{i\in I}$.

## Products and direct sums of modules

Let $I$ be a set and $\left({M}_{i}{\right)}_{i\in I}$ a family of $R-$modules. The product $\prod _{i\in I}{M}_{i}$ is the pair $∏ i∈I Mi ( πi )i∈I$ where

• $\prod _{i\in I}{M}_{i}$ is an $R-$module, and
• $\left({\pi }_{i}{\right)}_{i\in I}$ is a family of $R-$module homomorphisms ${\pi }_{i}:\prod _{i\in I}{M}_{i}\to {M}_{i}$
such that
1. if $P$ is an $R-$module and ${f}_{i}:P\to {M}_{i}$ is a family of $R-$module homomorphisms
2. then there is a unique $R-$module homomorphism $\stackrel{˜}{f}:P\to \prod _{i\in I}{M}_{i}$ such that
for all $i\in I$.

The direct sum $\underset{i\in I}{\oplus }{M}_{i}$ is the pair $⊕ i∈I Mi ( ιi ) i∈I$ where

• $\underset{i\in I}{\oplus }{M}_{i}$ is an $R-$module, and
• ${\iota }_{i}:{M}_{i}:\underset{i\in I}{\oplus }{M}_{i}$ is a family of $R-$module homomorphisms
such that
1. if $P$ is an $R-$module and ${\iota }_{i}:{M}_{i}\to \underset{i\in I}{\oplus }{M}_{i}$ is a family of morphisms
2. then there is a unique morphism $f:\underset{i\in I}{\oplus }{M}_{i}\to P$ such that
commutes for all $i$.

## Free groups

Let $X$ be a set. The free group on $X$ is the pair $X±1 ι$ where

• ${X}^{±1}$ is a group, and
• $\iota :X\to {X}^{±1}$ is a map
such that the pair $\left({X}^{±1},\iota \right)$ satisfies
1. if $G$ is a group, and $f:X\to G$ is a map
2. then there is a unique group homomorphism $\stackrel{˜}{f}:{X}^{±1}\to G$ such that
commutes.

#### Construction:

Define a group with set $X±1 = x i1 m1 ⋯ x ip mp x ij ∈ X, x ij ≠ x i j+1 and mi ∈ ℤ$ and with operation given by $( x 1 m1 ⋯ x p mp ) ( y 1 n1 ⋯ y q np ) = { x 1 m1 ⋯ x p-1 m p-1 x p mp + n1 y 2 n2 ⋯ y q nq , if xp = y1 x 1 m1 ⋯ x p mp y 1 n1 ⋯ y q np , otherwise. }$ Define $\iota :X\to {X}^{±1}$ by $\iota \left(x\right)={x}^{1}.$

$\left({X}^{±1},\iota \right)$ is the free group on $X$.

## Free modules

Let $X$ be a set. The free module with generating set $X$ is the pair $RX ι$ where

• ${R}^{X}$ is an $R-$module, and
• $\iota :X\to {R}^{X}$ is a map
such that
1. if $M$ is an $R-$module and $f:X\to M$ is a map
2. then there is a unique morphism $\stackrel{˜}{f}:{R}^{X}\to M$ such that
commutes.

#### Construction:

Define an $R-$module with set $R X = ∑ x∈X r x x r x ∈ R and all but finitely many r x = 0$ and operations Define $\iota :X\to {R}^{X}$ by $\iota \left(x\right)=x.$

$\left({R}^{X},\iota \right)$ is the free module on the set $X$.

## Tensor products

Let $M$ be a right $R-$module and $N$ a left $R-$module. The tensor product is the pair $\left(M{\otimes }_{R}N,\pi \right)$ where

• $M{\otimes }_{R}N$ is an abelian group, and
• $\pi :M×N\to M{\otimes }_{R}N$ is a $ℤ-$bilinear map
such that the pair $\left(M{\otimes }_{R}N,\pi \right)$ satisfies
1. if $P$ is an abelian group and $f:M×N\to P$ is a $ℤ-$bilinear map such that $f\left(mr,n\right)=f\left(m,rn\right)$
2. then there is a unique $ℤ-$bilinear map $\stackrel{˜}{f}:M{\otimes }_{R}N\to P$ such that
commutes.

#### Construction:

Define a $ℤ-$module with set $M ⊗R N = ∑ mi ⊗ ni mi ∈ M , ni ∈ N / ~$ where $~$ is the equivalence relation determined by $( m1 + m2 ) ⊗ n = m1 ⊗ n + m2 ⊗ n , m ⊗ ( n1 + n2 ) = m ⊗ n1 + m ⊗ n2 , and m r ⊗ n = m ⊗ r n ,$ for all $m\in M,\phantom{\rule{.5em}{0ex}}n\in N,\phantom{\rule{.5em}{0ex}}r\in R,$ and with operation $∑ mi ⊗ ni + ∑ mj ⊗ nj = ∑ i,j mi ⊗ ni + mj ⊗ nj .$

$\left(M{\otimes }_{R}N,\pi \right)$ is the tensor product of $M$ and $N$.

## Algebra of polynomials

Let $I$ be a set. The algebra of polynomials in the set $I$ is the free commutative associative algebra on $I$, i.e. the pair $R [ xi ] i∈I ι$ where

• $R\left[{x}_{i}{\right]}_{i\in I}$ is an $R-$algebra, and
• $\iota :I\to R\left[{x}_{i}{\right]}_{i\in I}$
and such that
1. if $S$ is an $R-$algebra and $f:I\to S$ is a map
2. then there is a unique $R-$algebra homomorphism $\stackrel{˜}{f}:R\left[{x}_{i}{\right]}_{i\in I}\to S$ such that
commutes.

## Group algebras

Let $G$ be a group. The group algebra of $G$ (over $R$) is the pair $R (G) ι$ where

• ${R}^{\left(G\right)}$ is an $R-$algebra, and
• $\iota :G\to {R}^{\left(G\right)}$ is a multiplication map
and the pair $\left({R}^{\left(G\right)},\iota \right)$ is such that
1. if $F$ is an $R-$algebra and $f:G\to F$ is a multiplicative map
2. then there is a unique $\stackrel{˜}{f}:{R}^{\left(G\right)}\to F$ such that
commutes.

## Tensor algebras

The tensor algebra of $M$ is the pair $T ( M ) ι$ where

• $T\left(M\right)$ is an $R-$algebra, and
• $\iota :M\to T\left(M\right)$ is an $R-$module homomorphism,
and the pair $\left(T\left(M\right),\iota \right)$ is such that
1. if $E$ is an $R-$algebra and $f:M\to E$ is an $R-$module homomorphism
2. then there is a unique $R-$algebra homomorphism $\stackrel{˜}{f}:T\left(M\right)\to E$ such that
commutes.

## Free associative algebras

Let $X$ be a set. The free associative algebra is the pair $R [ X* ] ι$ where

• $R\left[{X}^{*}\right]$ is an $R-$algebra, and
• $\iota :X\to R\left[{X}^{*}\right]$
such that
1. if $F$ is an $R-$algebra and $f:X\to F$
2. then there is a unique $R-$algebra homomorphism $\stackrel{˜}{f}:R\left[{X}^{*}\right]\to F$ such that
commutes.

## Notes and References

These notes are from lecture notes of Arun Ram from 2001.

## References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.