book:chap2:2.3_groups
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book:chap2:2.3_groups [2023/10/22 12:33] – [2.3 Groups] jv | book:chap2:2.3_groups [2023/10/22 12:53] (current) – jv | ||
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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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==== 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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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.1697970832.txt.gz · Last modified: 2023/10/22 12:33 by jv