Question

1. List all the cyclic subgroups

(a) ${\mathbf{U}}_{15}$

Short answer

The cyclic subgroups of${\mathbf{U}}_{15}$ are $⟨1⟩,⟨2⟩,⟨4⟩,⟨7⟩,⟨11⟩,⟨14⟩$.

Step-by-step solution

Cyclic Subgroup

Every element x in a group G, form a subgroup of all integer powers$⟨\mathrm{x}⟩=\{{\mathrm{x}}^{\mathrm{k}},\mathrm{k}\in \mathrm{\mathbb{Z}}\}$ is called a cyclic subgroupof x.

Elements in U15

The set of elements in${\mathrm{U}}_{15}$ is ${\mathrm{U}}_{15}=\{1,2,4,7,8,11,13,14\}$.

Cyclic Subgroups of U15

For element $1\in {\mathrm{U}}_{15}$,$1^1=1$and for any integer $\mathrm{k}\in \mathrm{\mathbb{Z}}$, $1^{\mathrm{k}}=1$. Thus, the cyclic subgroup generated by element 1 is $⟨1⟩=\left\{1\right\}$.

For element $2\in {\mathrm{U}}_{15}$, $2^1=2,2^2=4,2^3=8,2^4=16=1$. Thus, the cyclic subgroup generated by element 2 is $⟨2⟩=\{2,4,8,1\}$.

For element $4\in {\mathrm{U}}_{15}$,role="math" localid="1654320476796" $4^1=4,4^2=16=1$. Thus, the cyclic subgroup generated by element 4 isrole="math" localid="1654320473204" $⟨4⟩=\{4,1\}$.

For element $8\in U_{15}$, role="math" localid="1654320554971" $8^1=8,8^2=64=4,8^3=2,8^4=1$. Thus, the cyclic subgroup generated by element 7 is role="math" localid="1654320551623" $⟨8⟩=\{8,4,2,1\}=⟨2⟩$.

For element $11\in {\mathrm{U}}_{15}$, ${11}^1=11,{11}^2=121=1$. Thus, the cyclic subgroup generated by element 11 is $⟨11⟩=\{11,1\}$.

For element $13\in {\mathrm{U}}_{15}$, role="math" localid="1654320698314" ${13}^1=13,{13}^2=169=4,{13}^3=7,{13}^4=1$. Thus, the cyclic subgroup generated by element 13 is role="math" localid="1654320702679" $⟨13⟩=\{13,4,7,1\}=⟨7⟩$.

For element $14\in {\mathrm{U}}_{15}$, ${14}^1=14,{14}^2=196=1$. Thus, the cyclic subgroup generated by element 14 is $⟨14⟩=\{14,1\}$.

Therefore, there are 6 different cyclic subgroups which are $⟨1⟩,⟨2⟩,⟨4⟩,⟨7⟩,⟨11⟩,⟨14⟩$.