book:chap2:2.3_groups
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book:chap2:2.3_groups [2021/10/25 19:37] – [2.3.3 Self Test] jv | book:chap2:2.3_groups [2023/10/22 12:53] (current) – jv | ||
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+ | [[forcestorques|2. Balancing Forces and Torques]] | ||
+ | * [[ 2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]] | ||
+ | * [[ 2.2 Sets| 2.2 Sets ]] | ||
+ | * ** 2.3 Groups ** | ||
+ | * [[ 2.4 Fields| 2.4 Fields ]] | ||
+ | * [[ 2.5 Vector spaces| 2.5 Vector spaces ]] | ||
+ | * [[ 2.6 Physics application balancing forces| 2.6. Physics application: | ||
+ | * [[ 2.7 The inner product | 2.7 The inner product]] | ||
+ | * [[ 2.8 Cartesian coordinates | 2.8 Cartesian coordinates]] | ||
+ | * [[ 2.9 Cross products --- torques| 2.9 Cross products — torques ]] | ||
+ | * [[ 2.10 Worked example Calder' | ||
+ | * [[ 2.11 Problems| 2.11 Problems ]] | ||
+ | * [[ 2.12 Further reading| 2.12 Further reading ]] | ||
+ | |||
+ | ---- | ||
+ | |||
====== 2.3 Groups ====== | ====== 2.3 Groups ====== | ||
A group $G$ refers to a set of operations $t\in G$ that are changing some data or objects. | A group $G$ refers to a set of operations $t\in G$ that are changing some data or objects. | ||
Elementary examples refer to reflections in space, turning some sides of a Rubik' | Elementary examples refer to reflections in space, turning some sides of a Rubik' | ||
- | or translations in space, as illustrated in \cref{figure:middlePage}. | + | or translations in space, as illustrated in [[book:chap2: |
The subsequent action of two group elements $t_1$ and $t_2$ of $G$ | The subsequent action of two group elements $t_1$ and $t_2$ of $G$ | ||
is another (typically more complicated) transformation $t_3 \in G$. | is another (typically more complicated) transformation $t_3 \in G$. | ||
Line 10: | Line 26: | ||
The set of transformations forms a group iff it obeys the following rules. | The set of transformations forms a group iff it obeys the following rules. | ||
- | <WRAP box round> | + | <WRAP box round # |
A set $(G, \circ)$ is called a //group// with operation $\circ: G \times G \to G$ | A set $(G, \circ)$ is called a //group// with operation $\circ: G \times G \to G$ | ||
when the following rules apply | when the following rules apply | ||
Line 16: | Line 32: | ||
* **a) ** The set is // | * **a) ** The set is // | ||
* **b) ** The set has a //neutral element//: | * **b) ** The set has a //neutral element//: | ||
- | * **c) ** Each element has an //inverse element//: | + | * **c) ** Each element has an //inverse element//: |
* **d) ** The operation $\circ$ is // | * **d) ** The operation $\circ$ is // | ||
</ | </ | ||
Line 49: | Line 65: | ||
^ $1$ ^ $1$ | $0$ | | ^ $1$ ^ $1$ | $0$ | | ||
It describes the turning of a piece of paper: \\ | It describes the turning of a piece of paper: \\ | ||
- | Not turning, $0$, does not change anything (neutral element).\\ | + | - Not turning, $0$, does not change anything (neutral element).\\ |
- | Turning, $1$, shows the other side, and | + | - Turning, $1$, shows the other side, and turning twice is equivalent to not turning at all ($1$ is its own inverse). |
- | turning twice is equivalent to not turning at all ($1$ is its own inverse). | + | |
</ | </ | ||
Line 59: | Line 74: | ||
dihedral group of order 6 | dihedral group of order 6 | ||
with six elements | with six elements | ||
- | that is discussed in \cref{quest: | + | that is discussed in [[# |
</ | </ | ||
- | <WRAP right third> | ||
- | {{Buch-Hinten-Rechts-Indexed.png}} | ||
- | {{Buch-Rechts-Hinten-Indexed.png}} | ||
- | |||
- | Rotation of a book by multiples of $\pi/2$ around three orthogonal axes. | ||
- | <wrap hide> | ||
- | </ | ||
<WRAP box round> | <WRAP box round> | ||
The rotation of an object in space is a group. | The rotation of an object in space is a group. | ||
In particular this holds for the $90^\circ$-rotations of an object around a vertical and a horizontal axis. | In particular this holds for the $90^\circ$-rotations of an object around a vertical and a horizontal axis. | ||
- | \cref{fig:BuchDrehen} illustrates that these rotations do not commute. | + | [[#BuchDrehen |
</ | </ | ||
+ | <WRAP 100% centeralign> | ||
+ | <wrap #BuchDrehen 80%> | ||
+ | {{ Buch-Hinten-Rechts-Indexed.png? | ||
+ | first horizontal, then vertical.\\ | ||
+ | |||
+ | {{ buch-rechts-hinten-indexed.png? | ||
+ | first vertical, then horizontal.\\ | ||
+ | |||
+ | Figure 2.6: Rotation of a book by $\pi/2$ around a vertical and a horizontal axes. \\ $\quad$ | ||
+ | </ | ||
+ | </ | ||
+ | |||
+ | /* | ||
+ | <WRAP box round # | ||
+ | We consider the text fields of a fixed length $n$ in an electronic form. | ||
+ | Then the operations \\ | ||
+ | ``Put the letter $L$ into position $\bigsqcup$ of the field'' | ||
+ | with $L \in \{\_,\, a,\, \dots , z,\, A,\, \dots , Z\}$ \\ | ||
+ | and $\bigsqcup \in \{1, \dots, n\}$ form a group. | ||
+ | Also in this case one can easily check that the order of the operations is relevant. | ||
+ | In the left and right column the same operations are preformed for a text field of length $n=4$: | ||
+ | \begin{align*} | ||
+ | |\_|\_|\_|\_| | ||
+ | \to | M|\_|\_|\_| | ||
+ | \to | M| a|\_|\_| | ||
+ | \to | M| a| t|\_| & \qquad\qquad\qquad | ||
+ | \to | M| a| t| h| & \qquad\qquad\qquad | ||
+ | \to | M| a| t| s| & \qquad\qquad\qquad | ||
+ | \to | M| a| y| s| & \qquad\qquad\qquad | ||
+ | \to | M| h| y| s| & \qquad\qquad\qquad | ||
+ | \to | P| h| y| s| & \qquad\qquad\qquad | ||
+ | \end{align*} | ||
+ | </ | ||
+ | */ | ||
<wrap lo> | <wrap lo> | ||
\\ | \\ | ||
- | | + | - Depending of the context the inverse element is denoted as $g^{-1}$ or as $-g$. This depends on whether the operation is considered a multiplication or rather an addition. In accordance with this choice the neutral element is denoted as $1$ or $0$. |
- | - The second property of groups, b) $\exists e \in G \; \forall g \in G: e \circ g = g$, implies that also $g \circ e = g$. The proof is provided as \cref{quest: | + | |
- | - When a group is not commutative then one must distinguish the left and right inverse. The condition $g \circ i = e$ does not imply $i \circ g = e$. However, there always is another element $j \in G$ such that $j \circ g = e$. An example is provided in \cref{quest: | + | - The second property of groups, b) $\exists e \in G \; \forall g \in G: e \circ g = g$, implies that also $g \circ e = g$. The proof is provided as [[# |
+ | |||
+ | - When a group is not commutative then one must distinguish the left and right inverse. The condition $g \circ i = e$ does not imply $i \circ g = e$. However, there always is another element $j \in G$ such that $j \circ g = e$. An example is provided in [[# | ||
</ | </ | ||
+ | |||
+ | ----- | ||
==== 2.3.3 Self Test ==== | ==== 2.3.3 Self Test ==== | ||
- | Problem 2.4: <wrap hide> | + | <WRAP # |
- | ** Checking group axioms ** | + | |
Which of the following sets are groups? | Which of the following sets are groups? | ||
- | <WRAP group> | + | |
- | <WRAP third column> | + | - $( \mathbb{Z}, + )$ |
- | * a) $\quad ( \mathbb{N}, + )$ | + | |
- | | + | |
+ | | ||
+ | | ||
</ | </ | ||
- | <WRAP third column> | ||
- | * c) $\quad ( \mathbb{Z}, \cdot )$ | ||
- | * d) $\quad ( \{+1, -1\}, \cdot )$ | ||
- | </ | ||
- | <WRAP third column> | ||
- | * $\quad$e) | ||
- | * ∗ f) $\quad ( \{ 1, \dots, 12 \}, \oplus )$ | ||
- | </ | ||
- | </ | ||
- | where $\oplus$ in f) refers to adding as we do it on a clock, \\ | ||
- | e.g. $10 \oplus 4 = 2$. | ||
- | ---- | ||
- | Problem 2.5: <wrap hide> | ||
- | ** The group with three elements ** | ||
+ | <wrap # | ||
Let $(\mathcal G, \circ)$ be a group with three elements $\{n, | Let $(\mathcal G, \circ)$ be a group with three elements $\{n, | ||
- | where $n$ is the neutral element. | + | where $n$ is the neutral element. |
- | - Show that there only is a single choice for the result of the group operations $a \circ b$ with $a,b \in \mathcal G$. \\ Provide the group table. | + | - Show that there only is a single choice for the result of the group operations $a \circ b$ with $a,b \in \mathcal G$. Provide the group table. |
- Verify that the group describes the rotations of an equilateral triangle that interchange the positions of the angles. | - Verify that the group describes the rotations of an equilateral triangle that interchange the positions of the angles. | ||
- | - Show that there is a bijective map $\mathrm{m}: | + | - Show that there is a bijective map $\mathrm{m}: |
- | \begin{align*} | + | \[ |
- | \forall a,b \in \mathcal G \: : \: a \circ b = \bigl( \mathrm{m}(a) + \mathrm{m}(b) \bigr) \mathrm{mod} 3 \, . | + | \forall a,b \in \mathcal G : \mathrm{m}(a \circ b) = \bigl( \mathrm{m}(a) + \mathrm{m}(b) \bigr) \mathrm{mod} 3 |
- | \end{align*} | + | \] |
- | We say that the group $\mathcal G$ is isomorphic to the natural numbers with addition | + | We say that the group $\mathcal G$ is isomorphic to the natural numbers with addition modulo 3.((The natural number modulo $n$ amount to $n$ classes that represent the remainder of the numbers after division by $n$. |
- | The natural number modulo $n$ amount to $n$ classes that represent the remainder of the numbers after division by $n$. | + | |
For instance, for the natural numbers modulo two the $0$ represents even numbers, and the $1$ odd numbers. | For instance, for the natural numbers modulo two the $0$ represents even numbers, and the $1$ odd numbers. | ||
- | Similarly, for the natural numbers modulo three the $0$ represents numbers that are divisible by three, | + | Similarly, for the natural numbers modulo three the $0$ represents numbers that are divisible by three, |
- | and for the sum of $2$ and $2$ modulo $3$ one obtains | + | and for the sum of $2$ and $2$ modulo $3$ one obtains |
- | )) | + | |
- | ---- | ||
- | Problem 2.6: <wrap hide> | ||
- | ** Symmetry group of rectangles ** | ||
- | A polygon has a symmetry with an associated symmetry operation $a$ | + | <wrap # |
- | when $a$ only interchanges the vertices of the polygon. | + | |
- | It does not alter the position. | + | A polygon has a symmetry with an associated symmetry operation $a$ when $a$ only interchanges the vertices of the polygon. It does not alter the position. To get a grip on this concept we consider the symmetry operations of a rectangle. |
- | To get a grip on this concept we consider the symmetry operations of a rectangle. | + | |
- Sketch how reflections with respect to a symmetry axis interchange the vertices of a rectangle. What happens when the reflections are repeatedly applied? | - Sketch how reflections with respect to a symmetry axis interchange the vertices of a rectangle. What happens when the reflections are repeatedly applied? | ||
- Show that the symmetry operations form a group with four elements. Provide a geometric interpretation for all group elements. | - Show that the symmetry operations form a group with four elements. Provide a geometric interpretation for all group elements. | ||
- | - Provide the group table. | + | - Provide the group table. |
- | ---- | + | <WRAP right third #DihedralD6 > |
+ | {{: | ||
+ | [[https:// | ||
- | <WRAP third right> | + | Figure 2.7: Reflections of equilateral triangle with respect to the three symmetry axes form a group with six elements; see [[# |
- | {{cayley_graph_of_s3_with_triangles__generators_a_b.png}} | + | \\ |
+ | **Watch out:** in this figure the operations are carried out from left to right: | ||
+ | $ab$ refers here to $b$ after $a$, in contrast to the convention introduced in the beginning of the present section. | ||
+ | </ | ||
- | [[https:// | ||
- | [[https:// | ||
- | Reflections of equilateral triangle with respect to the three symmetry axes form a group with six elements; | + | < |
- | see \cref{quest: | + | |
- | < | + | |
- | </WRAP> | + | |
+ | [[# | ||
- | Problem | + | |
- | ** Dihedral | + | - Work out the group table. |
- | \\ | + | |
- | \cref{fig: | + | |
- | All group elements | + | - The group can also be represented in terms of a reflection |
- | e.g. those denoted as $a$ and $b$ in the figure. | + | |
- | - Verify that the group properties, \Defi{Group}, | ||
- | - Work out the group table. | ||
- | - Verify by inspection that $e$ is the neutral element for operation from the right //and// from the left. | ||
- | - Verify that the group is //not// commutative, | ||
- | - The group can also be represented in terms of a reflection and the rotations described in \cref{quest: | ||
- | ---- | ||
- | Problem 2.8: <wrap hide> | + | <wrap # |
- | ** Uniqueness of the neutral element ** | + | |
- | \\ | + | |
- | Proof that the group axioms, \Defi{Group}, | + | |
- | imply that | + | |
- | $e \circ g = g$ implies that also $g \circ e = g$. | + | |
+ | Proof that the group axioms, [[# | ||
~~DISCUSSION|Questions, | ~~DISCUSSION|Questions, | ||
+ |
book/chap2/2.3_groups.1635183460.txt.gz · Last modified: 2021/10/25 19:37 by jv