Theorem Summaries

The purpose of this page is to allow the statements of the theorems to be easily browsed in a single location.

Theorem 0.22
Let $S$ be a nonempty set and let $\sim$ be an equivalence relation on $S$. Then $\sim$ yields a partition of $S$, where $\bar{a}=\{x\in S \ | \ x\sim a\}$.

Theorem 2.13
Function composition is associative.

Theorem 3.13
If a binary structure has an identity element, the identity element is unique.

Theorem 3.14
The identity elements of isomorphic groups map to each other.

Theorem 4.15
Left and right cancellation are valid operations within a group.

Theorem 4.16
In groups, linear equations in one unknown have unique solutions.

Theorem 4.17
The inverse of each element in a group is unique.

Theorem 5.14
Listing of conditions for a subset of a group to be a subgroup (no need to prove associativity).

Theorem 5.17
Let $G$ be a group and let $a \in G$. Then: $H= \{a^{n} \ | \ n \in \mathbb{Z}\}$ is a subgroup of $G$ and is the smallest subgroup of $G$ that contains $a$,
that is, every subgroup containing $a$ contains $H$.

Theorem 6.1
Every cyclic group is abelian.

Theorem 6.6
Any subgroup of a cyclic group is cyclic.

Corollary 6.7
The subgroups of $\mathbb{Z}$ under addition are precisely the groups $n \mathbb{Z}$ under addition for $n \in \mathbb{Z}$.

Theorem 6.10
All infinite cyclic groups are isomorphic to $\langle \mathbb{Z}, + \rangle$.

Theorem 6.14
Let $G$ be a cyclic group with $n$ elements and generated by $a$. Let $b \in G$ and let $b = a^{s}$. Then $b$ generates a cyclic subgroup $H$ of $G$ containing $n/d$ elements, where $d$ is the greatest common divisor of $n$ and $s$. Also, $\langle a ^{s} \rangle = \langle a ^{t} \rangle$ if and only if $gcd(s,n) = gcd(t,n)$.

Theorem 8.5
Let $A$ be a nonempty set, and let $S_A$ be the collection of all permutations of $A$. Then $S_A$ is a group under permutation multiplication.

Lemma 8.15
Let $G$ and $G'$ be groups and let $\phi:G \rightarrow G'$ be a one-to-one function such that $\phi(xy)=\phi(x)\phi(y)$ for all $x,y\in G$.Then $\phi[G]$ is a subgroup of $G'$ and $\phi$ provides an isomorphism of $G$ with $\phi[G]$.

Theorem 8.16
Every group is isomorphic to a group of permutations.

Theorem 9.8
Every permutation $\sigma$ of a finite set is a product of disjoint cycles.

Theorem 9.15
No permutation in $S_n$ can be expressed both as a product of an even number of transpositions and as a product of an odd number of transpositions.

Theorem 9.20
If $n\geq 2$, then the collection of all even permutations of $\{1,2,3,...,n\}$ forms a subgroup of order $n!/2$ of the symmetric group $S_{n}$.

Theorem 10.1

Theorem 10.10
Let $H$ be a subgroup of a finite group $G$. Then the order of $H$ is a divisor of the order of $G$.

Lemma (Property of Cosets)

Theorem 10.12
The order of an element of a finite group divides the order of the group.

Theorem 10.14
Suppose $H$ and $K$ are subgroups of a group $G$ such that $K \leq H \leq G$, and suppose $(H:K)$ and $(G:H)$ are both finite. Then $(G:K)$ is finite, and $(G:K) = (G:H)(H:K)$.

Theorem 11.2
Let $G_{1},G_{2},...,G_{n}$ be groups. For $(a_{1},a_{2},...,a_{n})$ and $(b_{1},b_{2},...,b_{n})$ in $\prod_{i=1} ^{n} G_{i}$, define $(a_{1},a_{2},...,a_{n})(b_{1},b_{2},...,b_{n})$ to be the element $(a_{1}b_{1},a_{2}b_{2},...,a_{n}b_{n})$. Then $\prod_{i=1} ^{n} G_{i}$ is a group, the direct product of the groups $G_{i}$, under the binary operation.

Theorem 11.5
The group $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic and is isomorphic to $\mathbb{Z}_{mn}$ if and only if $m$ and $n$ are relatively prime, that is, the gcd of $m$ and $n$ is $1$.

Theorem 11.6
The group Πni=1Zmi is cyclic and isomorphic to Zm1m2⋯mn if and only if the numbers mi for i=1,⋯,n are such that the gcd of any two of them is 1.

Theorem 11.9
Let $(a_1,a_2,\dots,a_n)\in\prod^n_{i=1} G_i$.If $a_i$ is of finite order $r_i$ in $G_i$, then the order of $(a_1,a_2,\dots,a_n)$ in $\prod^n_{i=1} G_i$ is equal to the least common multiple of all the $r_i$.

Theorem 11.12 (Fundamental Theorem of Finitely Generated Abelian Groups)
Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups in the form

$\mathbb{Z}_{p_{1}^{r_{1}}} \times \mathbb{Z}_{p_{2}^{r_{2}}} \times \dots \times \mathbb{Z}_{p_{n}^{r_{n}}} \times \mathbb{Z} \times \mathbb{Z} \times \dots \times \mathbb{Z}$

where $p_{i}$ are primes, not necessarily distinct, and the $r_{i}$ are positive integers. The direct product is unique except for possible rearrangement of the factors; that is, the number (*Betti number* of G) of factors $\mathbb{Z}$ is unique and the prime powers $(p_{i})^{r_{i}}$ are unique

Theorem 11.15
The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime.

Theorem 11.16
If $m$ divides the order of a finite abelian group G, then G has a subgroup of order $m$.

Theorem 11.17
If $m$ is a square free integer,that is, $m$ is not divisible by the square of any prime, then every abelian group of order $m$ is cyclic.

Theorem 13.12 (properties of homomorphisms)
Let $\phi : G\rightarrow G'$ be a homomorphism of groups and let $g\in G$. Then:

(1) $\phi (e)$ is the identity of $G'$

(2) $\phi (g^{n}) = [\phi(g)]^{n}$ for all $n\in\mathbb{Z}$

(3) If $a\in G$, then $\phi (a^{-1}) = \phi (a)^{-1}$

(4) If $|g|$ is finite, then $|\phi (g)|$ divides $|g|$

(5) If $H\leq G$, then $\phi [H]\leq G'$

(6) If $K'\leq G'$, then $\phi ^{-1} [K'] \leq G$

(7) $Ker(\phi) \leq G$

Theorem 13.15
Let $\phi: G\rightarrow G'$ be a group homomorphism, and let $H=Ker(\phi)$.Let $a\in G$.Then the set

(1)
\begin{align} \phi^{-1}[\{\phi(a)\}]=\{x\in G|\phi(x)=\phi(a)\} \end{align}

is the left coset $aH$ of $H$,and is also the right coset $Ha$ of $H$.Consequently,the two partitions of $G$ into left cosets and into right cosets of $H$ are the same.

Theorem 13.18
A group homomorphism $\phi: G \rightarrow G'$ is a one-to-one map if and only if Ker$(\phi) = \{e\}$.

Theorem 13.20
If $\phi : G \rightarrow G'$ is a group homomorphism, then $Ker(\phi)$ is a normal subgroup of $G$.

Theorem 14.4
Let $H$ be a subgroup of a group $G$.Then left coset multiplication is well defined by the equation

(2)
$$(aH)(bH)=(ab)H$$

if and only if $H$ is a normal subgroup of $G$.

Theorem 14.5
Let H be a normal subgroup of G. Then the cosets of H form a group $G/H$ under the binary operation $(aH)(bH)=(ab)H$.

Theorem 14.9
Let $H$ be a normal subgroup of $G$. Then $\gamma :G\rightarrow G/H$ given by $\gamma (x)=xH$ is a homomorphism with kernel $H$.

Theorem 14.11 (The Fundamental Homomorphism Theorem)
Let $\phi : G \rightarrow G'$ be a group homomorphism with kernel $H$. Then $\phi[G]$ is a group, and $\mu : G/H \rightarrow \phi[G]$ given by $\mu(gH) = \phi(g)$ is an isomorphism. If $\gamma : G \rightarrow G/H$ is the homomorphism given by $\gamma(g) = gH$, then $\phi(g) = \mu \gamma(g)$ for each $g \in G$.

Theorem 14.13
The following are three equivalent conditions for a subgroup $H$ of a group $G$ to be a normal subgroup of $G$.

1. $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.
2. $gHg^{-1} = H$ for all $g \in G$.
3. $gH = Hg$ for all $g \in G$.

Condition (2) of this theorem is often taken as the definition of a normal subgroup $H$ of a group $G$.

Theorem 15.8
Let $G=H \times K$ be the direct product of groups $H$ and $K$. Then $\bar{H} = \{(h,e) | h \in H \}$ is a normal subgroup of $G$. Also $G/\bar{H}$ is isomorphic to $K$ in a natural way. Similarly, $G/\bar{K} \simeq H$ in a natural way.

Theorem 15.9
A factor of a cyclic group is cyclic

Theorem 15.15
The alternating group $A_{n}$ is simple for $n \geq 5$

Theorem 15.16
Let $\phi : G \rightarrow G'$ be a group homomorphism. If $N$ is a normal subgroup of $G$, then $\phi [N]$ is a normal subgroup of $\phi [G]$. Also, if $N'$ is a normal subgroup of $\phi [G]$, then $\phi ^{-1} [N']$ is a normal subgroup of $G$.

Theorem 15.18
$M$ is a maximal normal subgroup of $G$ if and only if $G/M$ is simple.

Theorem 15.20
Let $G$ be a group. The set of all commutators $aba^{-1}b^{-1}$ for $a,b \in G$ generates a subgroup $C$ (the commutator subgroup) of $G$. This subgroup $C$ is a normal subgroup of $G$. Furthermore, if $N$ is a normal subgroup of $G$, then $G/N$ is abelian if and only if $C \leq N$.

Theorem 14.13