Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.3_groups [2021/10/26 17:27] – abrilbook: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 ======
  
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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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 </wrap> </wrap>
  
- 
-<WRAP right third> 
-<wrap #BuchDrehen >  
-{{ Buch-Hinten-Rechts-Indexed.png?direct&200 |}} 
- 
-Figure 2.6: Rotation of a book by multiples of $\pi/2$ around three orthogonal axes. \\ $\quad$ 
-</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> \\ 
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 </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> \\ <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. We consider the text fields of a fixed length $n$ in an electronic form.
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 \end{align*} \end{align*}
 </WRAP> </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 [[#quest_groupSelftest-UniqueNeutralElt | Problem 2.8]].\\+ - 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]].  - 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]].
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 ==== 2.3.3 Self Test ==== ==== 2.3.3 Self Test ====
  
-<wrap #groupSelftest-01 > Problem 2.4: ** Checking group axioms **\\+<WRAP #groupSelftest-01 > Problem 2.4: ** Checking group axioms **\\
 Which of the following sets are groups? Which of the following sets are groups?
  
-<WRAP> +   -  $( \mathbb{N}, + )$ 
-  -  $( \mathbb{N}, + )$ +   -  $( \mathbb{Z}, + )$ 
-  -  $( \mathbb{Z}, + )$ +   -  $( \mathbb{Z}, \cdot )$ 
-  -  $( \mathbb{Z}, \cdot )$ +   -  $( \{+1, -1\}, \cdot )$ 
-  -  $( \{+1, -1\}, \cdot )$ +   -  $( \{ 0 \}, + )$ 
-  -  $( \{ 0 \}, + )$ +   -  :!: $( \{ 1, \dots, 12 \}, \oplus )$ where $\oplus$ refers to adding as we do it on a clock, e.g. $10 \oplus 4 = 2$.
-  -  :!: $( \{ 1, \dots, 12 \}, \oplus )$+
 </WRAP> </WRAP>
- 
-where $\oplus$ in 6) revers to adding as we do it on a clock, e.g. $10 \oplus 4 = 2$.</wrap>\\ 
  
  
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   -  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: 
- +\[  
-<WRAP centeralign> $\forall a,b \in \mathcal G : a \circ b = \bigl( \mathrm{m}(a) + \mathrm{m}(b) \bigr) \mathrm{mod} 3 \,$\\ +\forall a,b \in \mathcal G : \mathrm{m}(a \circ b) = \bigl( \mathrm{m}(a) + \mathrm{m}(b) \bigr) \mathrm{mod} 3  
-</WRAP> +\] 
- +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 $n$.
-We say that the group $\mathcal G$ is isomorphic to the natural numbers with addition addition modulo 3.((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  $(2+2)\mathrm{mod} 3 = 4\mathrm{mod} 3 = 1$.))
  
  
  
-<wrap #groupSelftest-03 > Problem 2.6: ** Symmetry group of rectangles **</wrap>\\+<wrap #groupSelftest-03 > Problem 2.6: ** Symmetry group of rectangles **</wrap>
  
 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. 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.
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 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]]. 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]].
 +\\
 +**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> </WRAP>
  
  
-<wrap #problem_27 > Problem 2.7: ** Dihedral group of order $6$ **</wrap>\\+<wrap #problem_27 > Problem 2.7: ** Dihedral group of order $6$ **</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. [[#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.
  
--  Verify that the group properties, [[#defi_group |Definition 2.6]], together with the three additional requirements +   -  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, \[ \mathcal G = \{ e, a, b, a\circ b, b\circ a, a\circ b\circ a \} \,. \] 
- +   - Work out the group table. 
-<WRAP centeralign> +   -  Verify by inspection that $e$ is the neutral element for operation from the right //and// from the left. 
-$a \circ a = b \circ b = e \quad\text{and}\quad a \circ b \circ a = b \circ a \circ b$ +   -  Verify that the group is //not// commutative, and provide an example of a group element where the left inverse and the right inverse differ. 
-</WRAP> +   -  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.
- +
-imply that the group has exactly six elements, +
- +
-<WRAP centeralign> +
-$\mathcal G = \{ e, a, b, a\circ b, b\circ a, a\circ b\circ a \} \,.$ +
-</WRAP>+
  
-- 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 [[#groupSelftest-02 |Problen 2.5]]. How would the graphical representation, analogous to [[#DihedralD6 |Figure 2.7]], look like in that case.\\ 
  
  
-<wrap #quest_groupSelftest-UniqueNeutralElt > Problem 2.8: ** Uniqueness of the neutral element **</wrap>\\+<wrap #quest_groupSelftest-UniqueNeutralElt > Problem 2.8: ** Uniqueness of the neutral element **</wrap>
  
-Proof that the group axioms, [[#defi_group |Definition 2.6]], 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.1635262029.txt.gz · Last modified: 2021/10/26 17:27 by abril