book:chap2:2.3_groups
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book:chap2:2.3_groups [2021/11/08 17:24] – external edit 127.0.0.1 | 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 ====== | ||
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* **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 // | ||
</ | </ | ||
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</ | </ | ||
- | |||
- | <WRAP right third> | ||
- | <wrap #BuchDrehen > | ||
- | {{ Buch-Hinten-Rechts-Indexed.png? | ||
- | |||
- | Figure 2.6: Rotation of a book by multiples of $\pi/2$ around three orthogonal axes. \\ $\quad$ | ||
- | </ | ||
- | </ | ||
<WRAP box round> | <WRAP box round> | ||
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</ | </ | ||
+ | <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 # | <WRAP box round # | ||
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 lo> | <wrap lo> | ||
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==== 2.3.3 Self Test ==== | ==== 2.3.3 Self Test ==== | ||
- | <wrap # | + | <WRAP # |
Which of the following sets are groups? | Which of the following sets are groups? | ||
- | < | + | - $( \mathbb{N}, + )$ |
- | | + | |
- | - $( \mathbb{Z}, + )$ | + | |
- | - $( \mathbb{Z}, \cdot )$ | + | |
- | - $( \{+1, -1\}, \cdot )$ | + | |
- | - $( \{ 0 \}, + )$ | + | |
- | - :!: $( \{ 1, \dots, 12 \}, \oplus )$ | + | |
</ | </ | ||
- | |||
- | where $\oplus$ in 6) revers to adding as we do it on a clock, e.g. $10 \oplus 4 = 2$.</ | ||
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- 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}: | ||
- | + | \[ | |
- | <WRAP centeralign> | + | \forall a,b \in \mathcal G : \mathrm{m}(a \circ b) = \bigl( \mathrm{m}(a) + \mathrm{m}(b) \bigr) \mathrm{mod} 3 |
- | </ | + | \] |
- | + | 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 $n$. | |
- | We say that the group $\mathcal G$ is isomorphic to the natural numbers with addition | + | |
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, | ||
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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 [[# | Figure 2.7: Reflections of equilateral triangle with respect to the three symmetry axes form a group with six elements; see [[# | ||
+ | \\ | ||
+ | **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. | ||
</ | </ | ||
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[[# | [[# | ||
- | - Verify that the group properties, [[# | + | - Verify that the group properties, [[# |
- | + | - Work out the group table. | |
- | <WRAP centeralign> | + | |
- | $a \circ a = b \circ b = e \quad\text{and}\quad a \circ b \circ a = b \circ a \circ b$ | + | - Verify that the group is //not// commutative, |
- | </ | + | |
- | + | ||
- | 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, | ||
- | - The group can also be represented in terms of a reflection and the rotations described in [[# | ||
book/chap2/2.3_groups.1636388659.txt.gz · Last modified: 2021/11/08 17:24 by 127.0.0.1