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Theorem 3.2

Let $\phi :G \rightarrow G'$ be a homomorphism of groups. Then,

\[ G/Ker \; \phi \cong Im \; \phi . \]

Hence, in particular, if $\phi $ is surjective then,
\[ G/Ker \; \phi \cong G'. \]

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Theorem 3.3

Let $H$ and $N$ be subgroups of $G$, and $N \vartriangleleft G$. Then,

\[ H/H \cap N \cong HN/N. \]

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Theorem 3.4

Let $H$ and $K$ be normal subgroups of $G$ and $K \subset H$. Then $H/K$ is a normal subgroup of $G/K$ and,

\[ (G/K)/(H/K) \cong G/H. \]

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Corollary 3.10

A Sylow $p$-subgroup of $G$ is unique if and only if it is normal.

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Corollary 3.11

Any $p$-subgroup of $G$ is contained in a Sylow $p$-subgroup.

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Lemma 5.16

$Z = \langle - I_L \rangle $.

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Lemma 5.17

If $p\neq 2$, then $L$ contains a unique element of order 2.

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Proposition 5.18

Each element of $L$ is conjugate to either $d_\delta $ for some $\delta \in F^\times $, or to $\pm s_\sigma $ for some $\lambda \in F$.

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Proposition 5.21

(i) $N_L(T_1) \subset H$, where $T_1$ is any subgroup of $T$ with order greater than 1.

(ii) $C_L(\pm s_\sigma ) = T \times Z$ where $\lambda \neq 0$.

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Proposition 5.22

(i) $N_L(D_1) = \langle D , w \rangle $, where $D_1$ is any subgroup of $D$ with order greater than 2.

(ii) $C_L(d_\delta )= D$ where $\delta \neq \pm 1$.

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Corollary 5.24

The centraliser of an element $x$ in $L$ is abelian unless $x$ belongs to the centre of $L$.

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Theorem 5.27

Let $\mathscr {L}$ be the projective line over the field $F$. Then $L$ is triply transitive on the set of the points of $\mathscr {L}$.

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Proposition 5.28

(i) Each element of the form $d_\delta $ (with $\delta \neq \pm 1$), fixes the same two points on the projective line $\mathscr {L}$ and fix no other point.
(ii) Each element of the form $\pm s_\sigma $ (with $\lambda \neq 0$), fixes the same point $P$ on $\mathscr {L}$ and fix no other point. Furthermore, Stab$(P) = H$.
(iii) All conjugate elements have the same number of fixed points on $\mathscr {L}$.
(iv) Any noncentral element of $L$ has at most 2 fixed points on $\mathscr {L}$.

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Theorem 6.6

Let $G$ be an arbitrary finite subgroup of $L$ containing $Z$.
(i) If $x \in G \! \setminus \! Z$ then we have $C_G(x) \in \mathfrak {M}$.
(ii) For any two distinct subgroups $A$ and $B$ of $\mathfrak {M}$, we have

\begin{align*} A \cap B = Z. \end{align*}

(iii) An element $A$ of $\mathfrak {M}$ is either a cyclic group whose order is relatively prime to $p$, or of the form $Q \times Z$ where $Q$ is an elementary abelian Sylow $p$-subgroup of $G$.
(iv) If $A \in \mathfrak {M}$ and $|A|$ is relatively prime to $p$, then we have $[N_G(A): A] \leq 2$. Furthermore, if $[N_G(A): A] = 2$, then there is an element $y$ of $N_G(A) \! \setminus \! A$ such that,

\begin{align*} yxy^{-1} = x^{-1} \qquad \forall x \in A.\end{align*}

(v) Let $Q$ be a Sylow $p$-subgroup of $G$. If $Q \neq \{ I_G\} $, then there is a cyclic subgroup $K$ of $G$ such that $N_G(Q) = QK$. If $|K| {\gt} |Z|$, then $K \in \mathfrak {M}$.

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Definition 3.6

Let $G$ be a finite group and $p$ a prime, a Sylow $\pmb {p}$-subgroup of $G$ is a subgroup of order $p^r$, where $p^{r+1}$ does not divide the order of $G$.

Let $p$ be a prime. A group $G$ is called a $\pmb {p}$-group if the order of each of it’s elements is a power of $p$. Similarly, a subgroup $H$ of $G$ is called a $\pmb {p}$-subgroup if the order of each of it’s elements is a power of $p$.

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Theorem 3.7

First Sylow Theorem. If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$.

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Theorem 3.8

All Sylow $p$-subgroups of G are conjugate.

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Theorem 3.9

The number of Sylow $p$-subgroups $n_p$ divides $m$ and satisfies $n_p \equiv 1 ($mod $p)$.

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Definition 3.12

Let $G$ be a group and $X$ be a set. Then $G$ is said to act on $X$ if there is a map $\phi : G \times X \rightarrow X$, with $\phi (a,x)$ denoted by $a^*x$, such that for $a,b \in G$ and $x \in X$, the following 2 properties hold:

\begin{align*} & (i) \quad a\, ^*(b\, ^*x) = (ab)^*x, \\ & (ii) \quad I_G\, ^*x = x. \end{align*}

The map $\phi $ is called the group action of $G$ on $X$.

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Definition 3.13

Let $G$ be a group acting on a set $X$ and let $x \in X$. Then the set,

\begin{align*} Stab(x) = \{ g \in G : gx = x \} , \end{align*}

is called the stabiliser of $x$ in $G$. Each $g$ in $S_G(x)$ is said to fix $x$, whilst $x$ is said to be a fixed point of each $g$ in $S_G(x)$. Also, the set,

\begin{align*} \text{Orb}(x) = \{ gx : g \in G \} , \end{align*}

is called the orbit of $x$ in $G$.

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Theorem 3.14

Let $G$ be a finite group acting on a set $X$. Then for each $x \in X$,

\[ |G| = |\text{Orb}(x)| |\text{Stab}(x)|. \]

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Definition 3.16

Let $G$ be a group and $a$ an element of $G$. An element $b \in G$ is said to be conjugate to $a$ if $b=xax^{-1}$ for some $x \in G$.

Let $H_1$ be a proper subgroup of $G$ and fix $x \in G \setminus H_1$. The set $H_2 = \{ g \in G : g= xh_1x^{-1}$, $\forall h_1 \in H_1\} $ is said to be a conjugate subgroup of $H_1$. We write $H_2 = xH_1x^{-1}$. It is trivial to show that $H_2$ is a subgroup of $G$.

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Definition 3.18

An automorphism of a group $G$ is a isomorphism from $G$ onto itself. The set of all automorphisms of $G$ forms a group under composition and is denoted by $Aut(G)$.

An inner automorphism is an automorphism whereby $G$ acts on itself by conjugation. That is, each $g \in G$ induces a map, $i_g : G \rightarrow G$, where $i_g(x) = g x g^{-1}$ for each $x \in G$. The set of all inner automorphisms is denoted by $Inn(G)$ and is a normal subgroup of $Aut(G)$ (For proof of this see [ 2 , p.104 ] .

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Definition 3.19

If $G_1, G_2,...,G_n$ are groups, we define a coordinate operation on the Cartesian product $G_1 \times G_2 \times ...\times G_n$ as follows:

\begin{align*} (a_1, a_2, ..., a_n) (b_1, b_2, ..., b_n) = (a_1 b_1, a_2 b_2, ..., a_n b_n), \end{align*}

where $a_i, b_i \in G_i$. It is easy to verify that $G_1 \times G_2 \times ...\times G_n$ is a group under this operation. This group is called the direct product of $G_1, G_2,...,G_n$.

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Lemma 5.14

$H / T \cong D$.

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Definition 5.15

The centre $Z(G)$ of a group $G$ is the set of elements of $G$ that commute with every element of $G$.

\begin{equation*} Z(G) = \{ z \in G : \forall g \in G, \hspace{6pt} gz=zg \} . \end{equation*}

It is an immediate observation that $Z(G)$ is a normal subgroup of $G$, since for each $z \in Z$, $gzg^{-1} = gg^{-1}z = z$, $\forall g \in G$. It’s also clear that a group is abelian if and only if $Z(G)=G$.

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Definition 5.19

The centraliser $C_G(H)$ of a subset $H$ of a group $G$ is the set of elements of $G$ which commute with each element of $H$.

\begin{equation*} C_G(H) = \{ g \in G : gh=hg, \quad \forall h\in H \} . \end{equation*}

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Definition 5.20

The normaliser $N_G(H)$ of a subset $H$ of a group $G$ is the set of elements of $G$ which stabilise $H$ under conjugation.

\begin{equation*} N_G(H) = \{ g \in G : gHg^{-1}=H\} . \end{equation*}

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Definition 5.25

Let $\mathscr {L}$ be the set of all 1-dimensional subspaces of $V$. A subset $\mathscr {S}$ of $\mathscr {L}$ is called a subspace of $\mathscr {L}$ if there is a subspace $U$ of $V$ such that $\mathscr {S}$ is the set of all 1-dimensional spaces of $U$. We have dim $U =$ dim $\mathscr {S} + 1$. The set $\mathscr {L}$ on which this concept of subspaces is defined is called the projective line on $V$ and an element of $\mathscr {L}$ is a 0-dimensional subspace of $\mathscr {L}$ and consequently called a point. The projective line can be considered as a straight line in the field, plus a point at infinity.

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Definition 5.26

Let $S$ be a permutation group which acts on a set $X$ and $\{ x_1, x_2, x_3 \} $ and $\{ x_1', x_2', x_3' \} $ be two subsets of distinct elements of $X$. Then $S$ is said be triply transitive on $X$ if there is an element $\pi \in S$ such that,

\begin{align*} x^{\pi }_i = x’_i, \qquad (\text{$i$ = 1,2 or 3}). \end{align*}

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Definition 6.7

The set $\mathcal{C}_i = \{ x A_i x^{-1} : x \in G \} $ is called the conjugacy class of $A_i \in \mathfrak {M}$.

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Definition 6.8

Let $A_i^*$ be the non-central part of $A_i \in \mathfrak {M}$, let $\mathfrak {M}^*$ be the set of all $A_i^*$ and let $\mathcal{C}_i^*$ be the conjugacy class of $A_i^*$.

For some $A_i \in \mathfrak {M}$ and $A_i^* \in \mathfrak {M}^*$ let,

\begin{align*} C_i = \bigcup \limits _{x \in G} x A_i x^{-1}, \quad \text{and} \quad C_i^* = \bigcup \limits _{x \in G} x A_i^* x^{-1}. \end{align*}

In other words, $C_i$ denotes the set of elements of $G$ which belong to some element of $\mathcal{C}_i$. It’s evident that $C_i^* = C_i \setminus Z$ and that there is a $C_i$ corresponding to each $\mathcal{C}_i$. Clearly we have the relation,

\begin{align} \label{orderorder} |C_i^*| = |A_i^*||\mathcal{C}_i^*|. \end{align}

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Lemma 4.8

The map $\bar{\varphi }$ is bijective

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Lemma 7.1

Let $H$ be a proper subgroup of a $p$-group $G$. Then $H \subsetneq N_G(H)$.

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Lemma 7.2

Let $Q$ be a Sylow $p$-subgroup and $K$ a maximal abelian subgroup of $G$ such that $N_G(Q) = QK$ and $Q \cap K = \{ I_G \} $. If $[N_G(K) : K] = 2$, then $Q$ is not a normal subgroup of $G$.

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Theorem 3.5

If the order of a finite group $G$ is divisible by a prime number $p$, then $G$ has an element of order $p$.

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Lemma 4.3

Let $R$ be a non-trivial commutative ring, then $Z(\operatorname{SL}_n(R)) \subseteq \ker (\varphi )$.

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Proposition 5.23

Let $a$ and $b$ be conjugate elements in a group $G$. Then $\exists \, x \in G$ such that $xC_G(a)x^{-1} = C_G(b)$.

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Proposition 3.17

Let $a$, $b$ be conjugate elements of a group $G$ and $A$, $B$ be conjugate subgroups of $G$. Then the following properites hold:
(i) If either $a$ or $b$ has finite order, then both $a$ and $b$ have the same order.
(ii) $A \cong B$.

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Lemma 3.20

Let $A$ and $B$ be normal subgroups of $G$ with $A \cap B = \{ I_G \} $. Then $AB \cong A \times B$.

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Lemma 3.21

Let $A$ and $B$ be subgroups of $G$. If $A \cap B = \{ I_G \} $ and $ab = ba$ $\forall a \in A$, $b \in B$. Then $AB \cong A \times B$.

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Lemma 4.6

If $F$ is an algebraically closed field then for every $G \in \operatorname{GL}_n(F)$ there exists a nonzero constant $\alpha \in F^\times $ and an element $S \in \operatorname{SL}_n(F)$ such that

\begin{equation*} G = \alpha \cdot S \end{equation*}

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Lemma 6.5

Every finite subgroup of a multiplicative group of a field is cyclic.

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Lemma 4.2

Let $R$ be a commutative ring, then $G \in GL_n(F)$ belongs to center of $\operatorname{GL}_n(R)$, $Z(\operatorname{GL}_n(R))$ if and only if $G = r \cdot I$ where $r \in R^\times $.

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Lemma 4.5

The homomorphism $\bar{\varphi }$ is injective.

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Definition 6.2Elementary Abelian

A group $G$ is said to be elementary abelian if it is abelian and every non-trivial element has order $p$, where $p$ is prime.

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Definition 6.1Maximal Abelian Subgroup

Let $H$ and $J$ be subgroups of a group $G$ where $H$ is abelian. $H$ is called maximal abelian if $J$ is not abelian whenever $H \subsetneq J$.

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Theorem 3.1

Let $G$ be a finite group. Then the order of any subgroup of $G$ divides the order of $G$.

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Theorem 7.6

Let $F$ be an arbitary algebraically closed field of characteristic $p$. Any finite subgroup $G$ of $\operatorname{SL}_2(F)$ is isomorphic to one of the following groups.
Class I: When $p=0$ or $|G|$ is relatively prime to $p$:
(i) A cyclic group.
(ii) The group defined by the presentation:

\begin{equation*} \langle \, x,y \, | \, x^n = y^2, \, yxy^{-1} = x^{-1} \, \rangle . \end{equation*}

(iii) The Special Linear Group $\operatorname{SL}_2(3)$.
(iv) The Special Linear Group $\operatorname{SL}_2(5)$.
(v) $\widehat{S}_4$, the representation group of $S_4$ in which the transpositions correspond to the elements of order $4$.

Class II: When $|G|$ is divisible by $p$:
(vi) $Q$ is elementary abelian, $Q \vartriangleleft G$ and $G/Q$ is a cyclic group whose order is relatively prime to $p$.
(vii) $p=2$ and $G$ is a dihedral group of order $2n$, where $n$ is odd.
(viii) The Special Linear Group $\operatorname{SL}_2(5)$, where $p=3=q$.
(ix) The Special Linear Group $\operatorname{SL}_2(\mathbb {F}_q)$.
(x) The group $\langle \operatorname{SL}_2(\mathbb {F}_q), d_\pi \rangle $, where $\operatorname{SL}_2(\mathbb {F}_q) \vartriangleleft \langle \operatorname{SL}_2(\mathbb {F}_q), d_\pi \rangle $.

Here, $Q$ is a Sylow $p$-subgroup of $G$ of order $q$, $\mathbb {F}_q$ is a field of $q$ elements, $\mathbb {F}_{q^2}$ is a field of $q^2$ elements, $\pi \in \mathbb {F}_{q^2} \setminus \mathbb {F}_q$ and $\pi ^2 \in \mathbb {F}_q$.

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Definition 6.3

Let $\mathfrak {M}$ denote the set of all maximal abelian subgroups of $G$.

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Lemma 7.5

Let $N$ be a normal subgroup of a group $G$ and let $H$ be a subgroup of $G$ which contains $N$.Then,

\begin{align*} H / N \vartriangleleft G / N \iff H \vartriangleleft G \end{align*}

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Lemma 7.4

Let $\mathbb {F}_q$ be the field of $q$ elements, where $q$ is the power of a prime. The order of $GL(2,\mathbb {F}_q)$ is $(q^2-1)(q^2-q)$ and the order of $\operatorname{SL}_2(\mathbb {F}_q)$ is $q(q^2-1)$.

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Theorem 6.9

Let $G$ be a finite subgroup of $L$ and $S$ be a subset of $\mathfrak {M}^*$ containing exactly one element from each of its conjugacy classes.

(i) The set of $C_i^*$ form a partition of $G \! \setminus \! Z$. That is,

\begin{align*} G \! \setminus \! Z = \bigcup \limits _{A_i^* \in S} C_i^*, \qquad \text{and} \qquad C_i^* \cap C_j^* = \varnothing , \qquad \forall \; i \neq j. \end{align*}

(ii)   $|\mathcal{C}_i^*| = |\mathcal{C}_i|$.

(iii)   $|\mathcal{C}_i| = [G : N_G(A_i)]$.

(iv)

\[ |G \! \setminus \! Z| = \sum _{A_i^* \in S} |A_i^*| [G:N_G(A_i)]. \]

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Theorem 4.9

If $F$ is an algebraically closed field, then the map $\bar{\varphi } : \operatorname{PSL}_n(F) \rightarrow \operatorname{PGL}_n(F)$ defines a group isomorphism between $\operatorname{PSL}_n(F)$ and $\operatorname{PGL}_n(F)$.

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Lemma 6.4

If $G$ is a finite group of order $p^m$ where $p$ is prime and $m{\gt}0$, then $p$ divides $|Z(G)|$.

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Definition 4.4

Given $Z(\operatorname{SL}_n(F)) \ker \varphi $ as shown in 4.3, by the universal property there exists a unique homomorphism $\bar{\varphi } : \operatorname{PSL}_n(F) \rightarrow \operatorname{PGL}_n(F)$ which is the lift of $\varphi $. Where $\varphi = \bar{\varphi } \circ \pi _{Z(\operatorname{SL}_n(F))}$ and $\pi _{Z(\operatorname{SL}_n(F))} : \operatorname{SL}_n(F) \rightarrow \operatorname{PSL}_n(F)$ is the canonical homomorphism from the group into its quotient.

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Definition 4.1

Let $\varphi : \operatorname{SL}_n(R) \rightarrow \operatorname{PGL}_n(R)$ be the injection of $\operatorname{PSL}_n(R)$ into $\operatorname{PGL}_n(R)$ defined by

\[ S \mapsto i(S) \; (R^\times I) \]

where $i : \operatorname{SL}_n(F) \hookrightarrow \operatorname{GL}_n(F)$ is the natural injection of the special linear group into the general linear group.

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Definition 5.2The diagonal matrix of $SL_2(F)$

Given an element $\delta \in F$ we define the shear matrix:

\[ s_\sigma = \begin{pmatrix} 1 & 0 \\ \sigma & 1 \end{pmatrix} \]

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Lemma 5.4

For any $\delta , \rho \in F^\times $ we have that

\[ d_\delta d_\rho = d_{\delta \rho } \]

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Lemma 5.6

We have that for all $\delta \in F^\times $ and $\sigma \in F$

\[ d_\delta s_\sigma d^{-1}_\delta = s_{\sigma \delta ^{-2}}. \]

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Definition 5.1The shear matrix of $SL_2(F)$

Given an element $\delta \in F^\times $ we define the diagonal matrix:

\[ d_\delta = \begin{pmatrix} \delta & 0 \\ 0 & \delta ^{-1} \end{pmatrix} \]

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Lemma 5.5

For any $\sigma , \gamma \in F$ we have that

\[ s_\sigma s_\gamma = s_{\sigma + \gamma }. \]

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Definition 5.3Rotation by $\pi / 2$ radians matrix

We denote the matrix which corresponds to a rotation by $\pi / 2$ radians to be:

\[ w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

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Lemma 5.7

For any $\delta \in F^\times $ we have:

\[ w d_\delta w^{-1} = d^{-1}_\delta . \]

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Definition 5.8The subgroup of diagonal matrices

The set of diagonal matrices with matrix multiplication is a subgroup of $\operatorname{SL}_2(F)$:

\[ D = \{ d_\delta \} = \left\{ \begin{pmatrix} \delta & 0 \\ 0 & \delta ^{-1} \end{pmatrix} \; | \; \delta \in F^\times \right\} \]

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Lemma 5.11$(D, \cdot ) \cong (F^\times , \cdot )$

The map $\phi : F^\times \overset {\sim }{\rightarrow } D$ defined by $d_\delta \mapsto (d_\delta )_{11}$ defines a group isomorphism.

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Definition 5.10The subgroup of lower triangular matrices

The set of lower triangular matrices with matrix multiplication is a subgroup of $\operatorname{SL}_2(F)$

\[ L = DS \]

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Lemma 5.13

$S$ is a normal subgroup of $H$

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Definition 5.9The subgroup of shear matrices

The set of shear matrices with matrix multiplication is a subgroup of $\operatorname{SL}_2(F)$:

\[ S = \{ s_\sigma \} = \left\{ \begin{pmatrix} 1 & 0 \\ \sigma & 1 \end{pmatrix} \; | \; \sigma \in F \right\} \]

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Lemma 5.12$(S, \cdot ) \cong (F, +)$

The map $\phi : F \overset {\sim }{\rightarrow } S$ defined by $s_\sigma \mapsto (s_\sigma )_{11}$ defines a group isomorphism.

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Lemma 7.3

Let $p$ be the prime characteristic of $F$ and let $q= p^k$ for some $k{\gt}0$. Set,

\begin{align} \label{RRR} R = \{ \lambda \in F : \lambda ^q -\lambda = 0 \} . \end{align}

Then $R$ is a subfield of $F$.

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Lemma 4.7

The map $\bar{\varphi }$ is surjective.

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Theorem 3.15

Let $G$ be a group and $H$ a subgroup of $G$ of finite index $n$. Then there is a homomorphism $\phi : G \longrightarrow S_n$ such that,

\begin{align*} ker(\phi ) = \bigcap \limits _{x \in G} x H x^{-1}. \end{align*}

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Lemma 6.11

$N_G(Q \times Z) = N_G(Q)$.

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Lemma 6.10

$N_G(A) =N_G(A^*)$.

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