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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
@@ Line 32: / Line 32: @@
   * **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//:  $\quad \exists e \in G \; \forall g \in G: e \circ g = g$.
-  * **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//:   $\quad \forall g_1, g_2, g_3 \in G : ( g_1 \circ g_2 ) \circ g_3 = g_1 \circ ( g_2 \circ g_3 )$.
 </WRAP>
@@ Line 132: / Line 132: @@
 ==== 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?
-<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>
-where $\oplus$ in 6) revers to adding as we do it on a clock, e.g. $10 \oplus 4 = 2$.</wrap>
@@ Line 178: / Line 175: @@
 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>
@@ Line 185: / Line 185: @@
 [[#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.