• Syllabus

4.11

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Sub sections and their related questions

Subgroups, proper subgroups.

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\)...
• 10M.2.hl.TZ0.1b: The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and...
• 08M.1.hl.TZ0.2b: (i)     Determine the order of each element of \(\left\{ {G,\left.  *  \right\}} \right.\)...
• 08M.2.hl.TZ0.4B: Consider the group \(\left\{ {G, * } \right\}\) and let \(H\) be a subset of \(G\) defined...
• 12M.2.hl.TZ0.4B.c: Suppose now that \(m = ab\) where \(a\) , \(b\) are unequal prime numbers. Show that...
• SPNone.2.hl.TZ0.4d: (i)     Find the order of all the elements of \(G\) . (ii)     Write down all the proper...
• 14M.1.hl.TZ0.16: \(\{ G,{\text{ }} * \} \) is a group of order \(N\) and \(\{ H,{\text{ }} * \} \) is a proper...
• 15M.2.hl.TZ0.9d: Show that if \(S\) is a subgroup of \(G\), then \(f(S)\) is a subgroup of \(H\).

Use and proof of subgroup tests.

• 15M.2.hl.TZ0.9d: Show that if \(S\) is a subgroup of \(G\), then \(f(S)\) is a subgroup of \(H\).

Definition and examples of left and right cosets of a subgroup of a group.

None

Lagrange’s theorem.

None

Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

None