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8.9

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Topic 8 - Option: Sets, relations and groups » 8.9

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Directly related questions

18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...
18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).
16N.3srg.hl.TZ0.3b: (i) State a generator for \(\{ H,{\text{ }} * \} \). (ii) Write down the elements of...
16N.3srg.hl.TZ0.3a: State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an...
17N.3srg.hl.TZ0.1a.ii: State whether or not \(\{ G,{\text{ }}{ \times _{18}}\} \) is cyclic, justifying your answer.
17N.3srg.hl.TZ0.1a.i: Find the order of elements 5, 7 and 17 in \(\{ G,{\text{ }}{ \times _{18}}\} \).
17M.3srg.hl.TZ0.4b: Show that there is no element of order 2.
15N.3srg.hl.TZ0.4c: Find the order of each element in \(T\).
15N.3srg.hl.TZ0.3d: (i) Find the maximum possible order of an element in \(\{ H,{\text{ }} \circ \} \). (ii) ...
15N.3srg.hl.TZ0.3a: Find the order of \(\{ G,{\text{ }} \circ \} \).
12M.3srg.hl.TZ0.5a: (i) Show that \(gh{g^{ - 1}}\) has order 2 for all \(g \in G\). (ii) Deduce that gh = hg...
08M.3srg.hl.TZ1.3: (a) Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form...
08M.3srg.hl.TZ1.5: Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let G be the group...
08M.3srg.hl.TZ2.1: (a) Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
11M.3srg.hl.TZ0.1b: (i) Show that {S , \( * \)} is a group. (ii) Find the order of each element of {S ,...
09M.3srg.hl.TZ0.1: (a) Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b) ...
09N.3srg.hl.TZ0.5: Let {G , \( * \)} be a finite group of order n and let H be a non-empty subset of G . (a) ...
SPNone.3srg.hl.TZ0.3a: (i) Write down the Cayley table for \(\{ G,{\text{ }}{ \times _7}\} \) . (ii) Determine...
10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a) Prove that G is Abelian. (b) Given that a is a...
10N.3srg.hl.TZ0.4: Set...
13M.3srg.hl.TZ0.2d: Determine the order of each element of \(\{ G,{\text{ }}{ \times _{14}}\} \).
11N.3srg.hl.TZ0.1d: Show that \(\{ H,{\text{ }} * \} \) is not cyclic.
13N.3srg.hl.TZ0.2a: (i) Prove that \(G\) is cyclic and state two of its generators. (ii) Let \(H\) be the...
15M.3srg.hl.TZ0.1b: (i) State the inverse of each element. (ii) Determine the order of each element.
14N.3srg.hl.TZ0.1b: Find the order of each of the elements of the group.
14N.3srg.hl.TZ0.4b: Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a...

Sub sections and their related questions

The order of a group.

SPNone.3srg.hl.TZ0.3a: (i) Write down the Cayley table for \(\{ G,{\text{ }}{ \times _7}\} \) . (ii) Determine...
13M.3srg.hl.TZ0.2d: Determine the order of each element of \(\{ G,{\text{ }}{ \times _{14}}\} \).
11N.3srg.hl.TZ0.1d: Show that \(\{ H,{\text{ }} * \} \) is not cyclic.
13N.3srg.hl.TZ0.2a: (i) Prove that \(G\) is cyclic and state two of its generators. (ii) Let \(H\) be the...
15N.3srg.hl.TZ0.3a: Find the order of \(\{ G,{\text{ }} \circ \} \).
16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).
16N.3srg.hl.TZ0.3a: State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an...
16N.3srg.hl.TZ0.3b: (i) State a generator for \(\{ H,{\text{ }} * \} \). (ii) Write down the elements of...
18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

The order of a group element.

12M.3srg.hl.TZ0.5a: (i) Show that \(gh{g^{ - 1}}\) has order 2 for all \(g \in G\). (ii) Deduce that gh = hg...
08M.3srg.hl.TZ1.5: Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let G be the group...
11M.3srg.hl.TZ0.1b: (i) Show that {S , \( * \)} is a group. (ii) Find the order of each element of {S ,...
09N.3srg.hl.TZ0.5: Let {G , \( * \)} be a finite group of order n and let H be a non-empty subset of G . (a) ...
14N.3srg.hl.TZ0.1b: Find the order of each of the elements of the group.
15M.3srg.hl.TZ0.1b: (i) State the inverse of each element. (ii) Determine the order of each element.
15N.3srg.hl.TZ0.3d: (i) Find the maximum possible order of an element in \(\{ H,{\text{ }} \circ \} \). (ii) ...
15N.3srg.hl.TZ0.4c: Find the order of each element in \(T\).
16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).
16N.3srg.hl.TZ0.3a: State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an...
16N.3srg.hl.TZ0.3b: (i) State a generator for \(\{ H,{\text{ }} * \} \). (ii) Write down the elements of...
18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

Cyclic groups.

08M.3srg.hl.TZ1.3: (a) Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form...
08M.3srg.hl.TZ2.1: (a) Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
09M.3srg.hl.TZ0.1: (a) Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b) ...
10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a) Prove that G is Abelian. (b) Given that a is a...
10N.3srg.hl.TZ0.4: Set...
14N.3srg.hl.TZ0.4b: Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a...
16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).
16N.3srg.hl.TZ0.3a: State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an...
16N.3srg.hl.TZ0.3b: (i) State a generator for \(\{ H,{\text{ }} * \} \). (ii) Write down the elements of...
18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

Generators.

09M.3srg.hl.TZ0.1: (a) Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b) ...
10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a) Prove that G is Abelian. (b) Given that a is a...

Proof that all cyclic groups are Abelian.

10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a) Prove that G is Abelian. (b) Given that a is a...