Groups：練習問題

A finite group \( G \) has order 20. According to Lagrange's theorem, which of the following is not a possible order for a subgroup of \( G \)?

Which of the following pairs of groups is isomorphic?

In the group of units modulo 13 under multiplication, find the order of the element 3.

Let \( M \) be the set of \( 2 \times 2 \) matrices of the form \( \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} \) where \( n \in \mathbb{Z} \). Under the operation of matrix multiplication, which of the following statements about \( M \) is true?

Let \( G \) be a cyclic group of order 18. How many elements in \( G \) are generators of the group?

In the group \( (\mathbb{Z}_{14}, +) \), where the binary operation is addition modulo 14, determine the order of the element 6.

A finite group \( G \) has order 42. A subgroup \( H \) of \( G \) is known to contain an element of order 7. Use Lagrange’s theorem to determine all possible values for the order of subgroup \( H \).

Consider the set S = \{1, 3, 7, 9\} which forms a group under the operation of multiplication modulo 10. Identify the identity element of this group and find the inverse of the element 7.

A group \( S \) consists of the set \( \{1, 5, 7, 11\} \) under the operation of multiplication modulo 12.
(a) Construct a Cayley table for \( S \) to verify that the set is closed under this operation.
(b) Use your table to identify the identity element and state the inverse of each element in \( S \).
(c) Determine the order of each element in \( S \).
(d) List all the proper subgroups of \( S \).

Let \( G \) be a finite group with identity element \( e \).
(a) Prove that if \( x^2 = e \) for every element \( x \in G \), then \( G \) must be an abelian group.
(b) Consider the set of matrices \( M = \{ \mathbf{I}, \mathbf{A}, \mathbf{B}, \mathbf{C} \} \) under the operation of matrix multiplication, where:
\( \mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), \( \mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \), \( \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \), \( \mathbf{C} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \).
Show that \( M \) forms a group and state the order of each element.
(c) Use the orders of the elements to explain whether \( M \) is isomorphic to the cyclic group of order 4, \( C_4 \).
(d) State the name of the group structure that \( M \) represents.