Group Theory and Symmetries

Flashcards covering core group theory concepts including group definitions, subgroups, orders, standard group examples, and dihedral groups based on the lecture notes.

Binary Operation

A function $*: X \times X \rightarrow X$ mapping $(a, b) \mapsto a * b$ on a set $X$, requiring $X$ to be closed under $*$.

Group

A set $G$ equipped with a binary operation $\cdot$, denoted $(G, \cdot)$, that satisfies associativity, contains an identity element $e$, and contains an inverse $a^{-1}$ for every element $a \in G$.

Abelian Group

A group $(G, \cdot)$ whose operation is commutative, satisfying $a b = b a$ for all $a, b \in G$.

Dihedral Group ($D_n$)

The group of symmetries of a regular $n$-sided polygon, consisting of $n$ rotations and $n$ reflections, with a total of $2n$ elements.

Order of a Group

The total number of elements contained in a group $G$, denoted by $|G|$.

Order of an Element

The smallest positive integer $n$ such that $a^n = e$ for an element $a \in G$, denoted by $|a|$; if no such integer exists, the element has infinite order.

Subgroup

A subset $H$ of a group $(G, \cdot)$ that itself forms a group under the same binary operation, denoted $H \le G$.

Proper Subgroup

A subgroup $H$ of a group $G$ such that $H \neq G$, denoted by $H < G$.

Trivial Subgroup

The subgroup of a group $G$ consisting solely of the identity element, denoted $\{e\}$.

Subgroup Test Theorem

A theorem stating that a non-empty subset $H \subseteq G$ is a subgroup of $G$ if and only if $a b^{-1} \in H$ for all $a, b \in H$.

General Linear Group ($GL_n(\mathbb{R})$)

The set of all $n \times n$ invertible matrices with real entries, which forms a non-abelian group under matrix multiplication.

Division Algorithm

The theorem stating that given integers $a$ and $n > 0$, there exist unique integers $q$ and $r$ such that $a = n q + r$ with $0 \le r < n$.

Set of Integers Modulo $n$ ($\mathbb{Z}_n$)

The set $\{0, 1, 2, \dots, n-1\}$ representing all possible remainders when dividing an integer by $n$, forming a group under addition modulo $n$.

Invertible Elements Modulo $n$ ($U_n$)

The set of elements in $\mathbb{Z}_n$ that are relatively prime to $n$, forming a group under multiplication modulo $n$.

Uniqueness of Identity

A group proposition stating that if $e_1$ and $e_2$ are both identity elements in a group $G$, then $e_1 = e_2$.

Left and Right Cancellation Laws

Group properties where $a b = a c$ implies $b = c$ (left cancellation) and $a c = b c$ implies $a = b$ (right cancellation) for all $a, b, c \in G$.

Dihedral Group Algebraic Rules

The governing relations for $D_n$ given by $R^n = e$, $F^2 = e$, and $F R^{-1} = R F$, where $R$ represents a rotation and $F$ represents a reflection.

Product Subgroup ($HK$)

For an abelian group $G$ with subgroups $H$ and $K$, the set $HK = \{h k \mid h \in H, k \in K\}$, which is also a subgroup of $G$.

Binary Operation

A function $*: X \times X \rightarrow X$ mapping $(a, b) \rightarrow a * b$, requiring that the set $X$ is closed under the operation.

Group

A set $G$ equipped with a binary operation $\times$, denoted $(G, \times)$, satisfying associativity ($(ab)c = a(bc)$), identity ($\neq e \times G$ such that $a \times e = a = e \times a$), and inverses (for each $a \times G$, $\neq b \times G$ such that $ab = e = ba$).

Abelian Group

A group $(G, \times)$ that satisfies the commutative property, where $ab = ba$ for all $a, b \times G$.

Identity Uniqueness

The proposition stating that the identity element in a group $G$ is unique, proved by showing that if $e_1$ and $e_2$ are both identity elements, then $e_1 = e_1 e_2 = e_2$.

Cancellation Laws

Properties in a group stating that left cancellation ($ab = ac \rightarrow b = c$) and right cancellation ($ac = bc \rightarrow a = b$) hold for all $a, b, c \times G$.

Inverse Uniqueness

The proposition stating that for any element $a \times G$ in a group, its inverse is unique, denoted $a^{-1}$ (or $-a$ in additive notation).

Division Algorithm

Given integers $a$ and $n > 0$, there exist unique integers $q$ and $r$ such that $a = nq + r$ with $0 \times r < n$, where $r$ is the remainder when dividing $a$ by $n$.

Congruence Modulo $n$

The relation $a \times b \times \text{mod } n$, defined as $n$ divides $a - b$, meaning $a$ and $b$ share the same remainder when divided by $n$.

The Group $\times_n$

The set of possible remainders modulo $n$, $\times_n = \times 0, 1, 2, \times, n-1 \times$, which forms an abelian group under addition modulo $n$ with identity $0$ and inverse $n - a$ for element $a$.

The Group $U_n$

The set of invertible elements in $\times_n$ under multiplication modulo $n$, consisting of elements in $\times_n$ that are relatively prime to $n$.

General Linear Group $GL_n(\times)$

The non-abelian group under matrix multiplication consisting of all $n \times n$ invertible matrices with real entries.

Composition of Functions

An associative binary operation on functions $f, g, h: Y \rightarrow Y$, satisfying $((h \times g) \times f)(x) = h(g(f(x))) = (h \times (g \times f))(x)$.

Dihedral Group $D_n$

The group of symmetries of a regular $n$-sided polygon, containing $n$ rotations and $n$ reflections for a total of $2n$ elements.

Algebraic Rules of $D_n$

The defining relations for the dihedral group generated by rotation $R$ and reflection $F$: $R^n = e$, $F^2 = e$, $(RF)^2 = e$, and $FR^{-1} = RF$.

Order of a Group

The total number of elements in a group $G$, denoted $|G|$.

Order of an Element

The smallest positive integer $n$ such that $a^n = e$ for an element $a \times G$, denoted $|a|$; if no such positive integer exists, $a$ has infinite order.

Subgroup

A subset $H$ of a group $(G, \times)$, denoted $H \times G$, that is itself a group under the binary operation $\times$.

Proper Subgroup

A subgroup $H$ of $G$, denoted $H < G$, such that $H \times G$ and $H \neq G$.

Trivial Subgroup

The subgroup $\times e \times$ of a group $G$, consisting solely of the identity element.

Subgroup Test

A theorem stating that a nonempty subset $H \times G$ is a subgroup of $G$ if and only if $ab^{-1} \times H$ for every $a, b \times H$.

Product of Subgroups $HK$

For an abelian group $G$ with subgroups $H$ and $K$, the set $HK = \times hk \times h \times H, k \times K \times$, which is also a subgroup of $G$.

Finite Order Subgroup

In an abelian group $G$, the set $H = \times a \times G \times |a| \text{ is finite} \times$ consisting of all elements of finite order, which forms a subgroup of $G$.