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 10.10**

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

**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

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

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 $G$.

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