Groups Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Groups questions. See exactly how to solve problems on groups, modular-arithmetic, cyclic-groups, order-of-a-group.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The group G=(Z12,+12)G=\left(\mathbb{Z}_{12},+_{12}\right) is the set {0,1,2,…,11}\left\{0,1,2,\dots,11\right\} under addition modulo 1212. Write down the order of the group GG.

Worked solution

  1. List the elements of GG

    G={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}G=\left\{0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11\right\}

    Writing out the element set is the safest way to avoid missing or repeating an element.

  2. Count the elements

    ∣G∣=12\left|G\right|=12

    The order of a finite group is the number of elements in its element set.

  3. State the order of GG

    the order of G is 12\text{the order of}\ G\ \text{is}\ 12

    The order of a group is the number of elements it contains.

Answer
∣G∣=12\left|G\right|=12
Question 2
2 markseasy
The group G=(U9,×9)G=\left(U_{9},\times_{9}\right) has element set U9={1,2,4,5,7,8}U_{9}=\left\{1,2,4,5,7,8\right\}, the integers less than 99 that are coprime to 99, and the operation is multiplication modulo 99. Write down the order of the group GG.

Worked solution

  1. List the elements of GG

    G={1, 2, 4, 5, 7, 8}G=\left\{1,\ 2,\ 4,\ 5,\ 7,\ 8\right\}

    Writing out the element set is the safest way to avoid missing or repeating an element.

  2. Count the elements

    ∣G∣=6\left|G\right|=6

    The order of a finite group is the number of elements in its element set.

  3. Check that the identity has been included

    e=1∈Ge=1\in G

    Every group contains exactly one identity element.

  4. State the order of GG

    the order of G is 6\text{the order of}\ G\ \text{is}\ 6

    The order of a group is the number of elements it contains.

Answer
∣G∣=6\left|G\right|=6
Question 3
2 markseasy
The group G=(D4,∘)G=\left(D_{4},\circ\right) is the group of the 88 symmetries of a regular 4-gon under composition. Writing rr for the rotation through 2π4\frac{2\pi}{4} and ss for a reflection, G={e,r,…,r3,s,rs,…,r3s}G=\left\{e,r,\dots,r^{3},s,rs,\dots,r^{3}s\right\}, where r4=er^{4}=e, s2=es^{2}=e and sr=r3ssr=r^{3}s. Write down the order of the group GG.

Worked solution

  1. List the elements of GG

    G={e, r, r2, r3, s, rs, r2s, r3s}G=\left\{e,\ r,\ r^{2},\ r^{3},\ s,\ rs,\ r^{2}s,\ r^{3}s\right\}

    Writing out the element set is the safest way to avoid missing or repeating an element.

  2. Count the elements

    ∣G∣=8\left|G\right|=8

    The order of a finite group is the number of elements in its element set.

  3. State the order of GG

    the order of G is 8\text{the order of}\ G\ \text{is}\ 8

    The order of a group is the number of elements it contains.

Answer
∣G∣=8\left|G\right|=8
Question 4
2 markseasy
The set G={(1001), (0−110), (−100−1), (01−10)}G=\left\{\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},\ \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix},\ \begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix},\ \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\right\} of 2×22\times 2 matrices forms a group under matrix multiplication. Write down the order of the group GG.

Worked solution

  1. List the elements of GG

    G={(1001), (0−110), (−100−1), (01−10)}G=\left\{\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},\ \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix},\ \begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix},\ \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\right\}

    Writing out the element set is the safest way to avoid missing or repeating an element.

  2. Count the elements

    ∣G∣=4\left|G\right|=4

    The order of a finite group is the number of elements in its element set.

  3. Check that the identity has been included

    e=(1001)∈Ge=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\in G

    Every group contains exactly one identity element.

  4. State the order of GG

    the order of G is 4\text{the order of}\ G\ \text{is}\ 4

    The order of a group is the number of elements it contains.

Answer
∣G∣=4\left|G\right|=4
Question 5
2 markseasy
The group G=(Z12,+12)G=\left(\mathbb{Z}_{12},+_{12}\right) is the set {0,1,2,…,11}\left\{0,1,2,\dots,11\right\} under addition modulo 1212. Find the order of the element 33 of GG.

Worked solution

  1. Take successive powers of 33 until the identity appears

    3=3,32=6,33=9,34=03=3,\quad 3^{2}=6,\quad 3^{3}=9,\quad 3^{4}=0

    Each power is obtained by combining the previous one with 33 again.

  2. Identify the first power equal to the identity

    34=0=e3^{4}=0=e

    The order is the smallest positive index at which the identity is reached.

  3. State the order of the element

    ord⁡(3)=4\operatorname{ord}\left(3\right)=4

    This is the least positive integer power of 33 that gives the identity.

Answer
ord⁡(3)=4\operatorname{ord}\left(3\right)=4

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