Order, subgroups and Lagrange's theoremEdexcel A-Level Further Maths: Mind map

Exam questions on Order, subgroups and Lagrange's theorem

1. The set $G=\{1,2,3,\ldots,10\}$ under multiplication modulo 11 is a group.
   Find a subgroup of $G$ of order $5$, listing its elements.2 marks
2. $G$ is a finite group of order $24$.
   An element $g$ of $G$ has order $24$. Explain why $G$ must be cyclic.2 marks
3. The symmetries of an equilateral triangle form a group $T$ of order $6$ under composition. Its elements are the identity $e$, a rotation $r$ through $120^\circ$, the rotation $r^2$ through $240^\circ$, and three reflections $m_1,m_2,m_3$.
   State, with justification, the orders of $r$, $r^2$ and $m_1$.3 marks