Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.3_groups [2021/10/25 19:37] – [2.3.3 Self Test] jvbook: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: balancing forces]]
 +  * [[  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's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 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's cube, Elementary examples refer to reflections in space, turning some sides of a Rubik's cube,
-or translations in space, as illustrated in \cref{figure:middlePage}.+or translations in space, as illustrated in [[book:chap2:2.1_motivation_and_outline #fig_middlePage |Figure 2.1]].
 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 $t_3 \in G$.
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 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>**Definition 2.6** <wrap em>Group</wrap> \\ +<WRAP box round #defi_group >**Definition 2.6** <wrap em>Group</wrap> \\ 
 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
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   * **a) **  The set is //closed//:  $\quad \forall g_1, g_2 \in G : g_1 \circ g_2 \in G$.   * **a) **  The set is //closed//:  $\quad \forall g_1, g_2 \in G : g_1 \circ g_2 \in G$.
   * **b) **  The set has a //neutral element//:  $\quad \exists e \in G \; \forall g \in G: e \circ g = g$.   * **b) **  The set has a //neutral element//:  $\quad \exists e \in G \; \forall g \in G: e \circ g = g$.
-  * **c) **  Each element has an //inverse element//:   $\quad \forall g \in G \; \exists i \in G : g \circ i = e$.+  * **c) **  Each element has an //inverse element//:   $\quad \forall g \in G \; \exists i \in G : i \circ g = e$.
   * **d) **  The operation $\circ$ is //associative//:   $\quad \forall g_1, g_2, g_3 \in G : ( g_1 \circ g_2 ) \circ g_3 = g_1 \circ ( g_2 \circ g_3 )$.   * **d) **  The operation $\circ$ is //associative//:   $\quad \forall g_1, g_2, g_3 \in G : ( g_1 \circ g_2 ) \circ g_3 = g_1 \circ ( g_2 \circ g_3 )$.
 </WRAP> </WRAP>
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 ^ $1$      ^ $1$ | $0$ | ^ $1$      ^ $1$ | $0$ |
 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).+
 </WRAP> </WRAP>
  
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 dihedral group of order 6 dihedral group of order 6
 with six elements with six elements
-that is discussed in \cref{quest:groupSelftest-DihedralD6}. +that is discussed in [[#problem_27 |Problem 2.7]]. 
 </wrap> </wrap>
  
-<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>\label{fig:BuchDrehen}</wrap> 
-</WRAP> 
  
 <WRAP box round>**Example 2.9** <wrap em>Non-commutative groups: rotations</wrap> \\  <WRAP box round>**Example 2.9** <wrap em>Non-commutative groups: rotations</wrap> \\ 
 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 |Figure 2.6]] illustrates that these rotations do not commute.
 </WRAP> </WRAP>
  
 +<WRAP 100% centeralign>
 +<wrap #BuchDrehen 80%> 
 +{{ Buch-Hinten-Rechts-Indexed.png?direct&500 |}}
 +first horizontal, then vertical.\\
 +
 +{{ buch-rechts-hinten-indexed.png?direct&500 |}}
 +first vertical, then horizontal.\\
 +
 +Figure 2.6: Rotation of a book by $\pi/2$ around a vertical and a horizontal axes. \\ $\quad$
 +</wrap>
 +</WRAP>
 +
 +/*
 +<WRAP box round #Example_210>**Example 2.10** <wrap em>Non-commutative groups: edit text fields</wrap> \\
 +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*}
 +      |\_|\_|\_|\_|  & \qquad\qquad\qquad\quad \ \ |\_|\_|\_|\_| \\
 +  \to | M|\_|\_|\_|  & \qquad\qquad\qquad   \to | P|\_|\_|\_| \\
 +  \to | M| a|\_|\_|  & \qquad\qquad\qquad   \to | P| h|\_|\_| \\
 +  \to | M| a| t|\_|  & \qquad\qquad\qquad   \to | P| h| y|\_| \\
 +  \to | M| a| t| h|  & \qquad\qquad\qquad   \to | P| h| y| s| \\
 +  \to | M| a| t| s|  & \qquad\qquad\qquad   \to | P| h| y| h| \\
 +  \to | M| a| y| s|  & \qquad\qquad\qquad   \to | P| h| t| h| \\
 +  \to | M| h| y| s|  & \qquad\qquad\qquad   \to | P| a| t| h| \\
 +  \to | P| h| y| s|  & \qquad\qquad\qquad   \to | M| a| t| h|
 +\end{align*}
 +</WRAP>
 +*/
  
 <wrap lo>**Remark 2.7.**  Notations and additional properties: <wrap lo>**Remark 2.7.**  Notations and additional properties:
 \\ \\
-  -  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 $0$. + - 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:groupSelftest-UniqueNeutralElt}. + 
-  -  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:groupSelftest-DihedralD6}.+ - 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 [[#quest_groupSelftest-UniqueNeutralElt | Problem 2.8]]. 
 + 
 + - 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 [[#problem_27 |Problem 2.7]].
 </wrap> </wrap>
 +
 +-----
  
 ==== 2.3.3 Self Test ==== ==== 2.3.3 Self Test ====
  
-Problem 2.4: <wrap hide>\label{quest:groupSelftest-01}</wrap> +<WRAP #groupSelftest-01 > Problem 2.4: ** Checking group axioms **\\
-** Checking group axioms ** +
 Which of the following sets are groups? Which of the following sets are groups?
  
-<WRAP group> +   -  $( \mathbb{N}, + )$ 
-<WRAP third column> +   -  $( \mathbb{Z}, + )$ 
-  * a)  $\quad ( \mathbb{N}, + )$ +   -  $( \mathbb{Z}, \cdot )$ 
-  * b)  $\quad ( \mathbb{Z}, + )$+   -  $( \{+1, -1\}, \cdot )$ 
 +   -  $( \{ 0 \}, + )$ 
 +   -  :!: $( \{ 1, \dots, 12 \}, \oplus )$ where $\oplus$ refers to adding as we do it on a clock, e.g. $10 \oplus 4 = 2$.
 </WRAP> </WRAP>
-<WRAP third column> 
-  * c)  $\quad ( \mathbb{Z}, \cdot )$ 
-  * d)  $\quad ( \{+1, -1\}, \cdot )$ 
-</WRAP> 
-<WRAP third column> 
-  * $\quad$e)  $\quad ( \{ 0 \}, + )$ 
-  * ∗ f)      $\quad ( \{ 1, \dots, 12 \}, \oplus )$  
-</WRAP> 
-</WRAP> 
-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>\label{quest:groupSelftest-02}</wrap> 
-** The group with three elements ** 
  
 +<wrap #groupSelftest-02 > Problem 2.5: ** The group with three elements ** </wrap>\\
 Let $(\mathcal G, \circ)$ be a group with three elements $\{n,l,r\}$, Let $(\mathcal G, \circ)$ be a group with three elements $\{n,l,r\}$,
-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 $\mathrm{m}:\{n,l,r\} \to \{0,1,2\}$ with the following property:  +  -  Show that there is a bijective map $\mathrm{m}:\{n,l,r\} \to \{0,1,2\}$ with the following property: 
-\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 addition modulo 3.(( +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  $(2+2)\mathrm{mod} 3 = 4\mathrm{mod} 3 = 1$. +and for the sum of $2$ and $2$ modulo $3$ one obtains  $(2+2)\mathrm{mod} 3 = 4\mathrm{mod} 3 = 1$.))
-))+
  
----- 
  
-Problem 2.6: <wrap hide>\label{quest:groupSelftest-03}</wrap> 
-** Symmetry group of rectangles ** 
  
-A polygon has a symmetry with an associated symmetry operation $a$ +<wrap #groupSelftest-03 > Problem 2.6: ** Symmetry group of rectangles **</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 > 
 +{{:book:chap2:cayley_graph_of_s3_with_triangles_generators_a_b.png}} 
 +[[https://commons.wikimedia.org/wiki/File:Cayley_graph_of_S3_with_triangles;_generators_a,_b.svg|Watchduck (a.k.a. Tilman Piesk), wikimedia]], [[https://creativecommons.org/licenses/by-sa/4.0|CC BY-SA]]
  
-<WRAP third right> +Figure 2.7: Reflections of equilateral triangle with respect to the three symmetry axes form a group with six elements; see [[#problem_27 |Problem 2.7]]. 
-{{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. 
 +</WRAP>
  
-[[https://commons.wikimedia.org/wiki/File:Cayley_graph_of_S3_with_triangles;_generators_a,_b.svg|Watchduck (a.k.a. Tilman Piesk), wikimedia]] 
-[[https://creativecommons.org/licenses/by-sa/4.0|CC BY-SA]] 
  
-Reflections of equilateral triangle with respect to the three symmetry axes form a group with six elements; +<wrap #problem_27 > Problem 2.7: ** Dihedral group of order $6$ **</wrap>
-see \cref{quest:groupSelftest-DihedralD6}. +
-<wrap hide> \label{fig:DihedralD6} </wrap> +
-</WRAP>+
  
 +[[#DihedralD6 |Figure 2.7 ]] illustrates the effect of reflections of a triangle with respect to its three symmetry axis. All group elements can be generated by repeated action of two reflections, e.g. those denoted as $a$ and $b$ in the figure.
  
-Problem 2.7: <wrap hide>\label{quest:groupSelftest-DihedralD6}</wrap>  +   -  Verify that the group properties, [[#defi_group |Definition 2.6]], together with the three additional requirements \[ a \circ a = b \circ b = e \quad\text{and}\quad a \circ b \circ a = b \circ a \circ b \] imply that the group has exactly six elements, \[ \mathcal G = \{ e, a, b, a\circ b, b\circ a, a\circ b\circ a \} \,. \] 
-** Dihedral group of order $6$ ** +   - Work out the group table. 
-\\ +   -  Verify by inspection that $e$ is the neutral element for operation from the right //and// from the left. 
-\cref{fig:DihedralD6} illustrates the effect of reflections of a triangle with respect to its three symmetry axis. +   -  Verify that the group is //not// commutative, and provide an example of a group element where the left inverse and the right inverse differ. 
-All group elements can be generated by repeated action of two reflections, +   -  The group can also be represented in terms of a reflection and the rotations described in [[#groupSelftest-02 |Problen 2.5]]. How would the graphical representation, analogous to [[#DihedralD6 |Figure 2.7]], look like in that case.
-e.g. those denoted as $a$ and $b$ in the figure.+
  
-  -  Verify that the group properties, \Defi{Group}, together with the three additional requirements \\ $\quad a \circ a = b \circ b = e \quad $ and $\quad  a \circ b \circ a = b \circ a \circ b$  \\ imply that the group has exactly six elements, \\ $\quad \mathcal G = \{ e, a, b, a\circ b, b\circ a, a\circ b\circ a \} \,$. 
-  -  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, and provide an example of a group element where the left inverse and the right inverse differ. 
-  -  The group can also be represented in terms of a reflection and the rotations described in \cref{quest:groupSelftest-02}. How would the graphical representation, analogous to \cref{fig:DihedralD6}, look like in that case. 
  
----- 
  
-Problem 2.8: <wrap hide>\label{quest:groupSelftest-UniqueNeutralElt}</wrap>  +<wrap #quest_groupSelftest-UniqueNeutralElt > Problem 2.8: ** Uniqueness of the neutral element **</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, [[#defi_group |Definition 2.6]], imply that $e \circ g = g$ implies that also $g \circ e = g$.
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
 +
book/chap2/2.3_groups.1635183460.txt.gz · Last modified: 2021/10/25 19:37 by jv