Matrix Groups and Symmetry: exercises

Practice (19)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Prove the identity \[\begin{aligned}\end{aligned}\].

   جواب رو نشون بده

   Hint:

   \[\begin{aligned}\frac{1}{2} \left[ \|{\mathbf x} + {\mathbf y}\|^2 + \|{\mathbf x}\|^2 - \| {\mathbf y}\|^2 \right] & = \frac{1}{2} \left[ \langle x + y, x + y \rangle - \|{\mathbf x}\|^2 - \| {\mathbf y}\|^2 \right] \\ & = \frac{1}{2} \left[ \| {\mathbf x}\|^2 + 2 \langle x, y \rangle + \| {\mathbf y}\|^2 - \|{\mathbf x}\|^2 - \| {\mathbf y}\|^2 \right] \\ & = \langle {\mathbf x}, {\mathbf y} \rangle\end{aligned}\].

2. Show that \(O(n)\) is a group.

3. Prove that the following matrices are orthogonal. Are any of these matrices in \(SO(n)\)?

   1. \[\begin{aligned}\end{aligned}\]

   2. \[\begin{aligned}\end{aligned}\]

   3. \[\begin{aligned}\end{aligned}\]

   4. \[\begin{aligned}\end{aligned}\]

   جواب رو نشون بده

   Hint:

   (a) is in \(SO(2)\); (c) is not in \(O(3)\).

4. Determine the symmetry group of each of the figures below.

5. Let \({\mathbf x}\), \({\mathbf y}\), and \({\mathbf w}\) be vectors in \({\mathbb R}^n\) and \(\alpha \in {\mathbb R}\). Prove each of the following properties of inner products.

   1. \(\langle {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf y}, {\mathbf x} \rangle\).

   2. \(\langle {\mathbf x}, {\mathbf y} + {\mathbf w} \rangle = \langle {\mathbf x}, {\mathbf y} \rangle + \langle {\mathbf x}, {\mathbf w} \rangle\).

   3. \(\langle \alpha {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf x}, \alpha {\mathbf y} \rangle = \alpha \langle {\mathbf x}, {\mathbf y} \rangle\).

   4. \(\langle {\mathbf x}, {\mathbf x} \rangle \geq 0\) with equality exactly when \({\mathbf x} = 0\).

   5. If \(\langle {\mathbf x}, {\mathbf y} \rangle = 0\) for all \({\mathbf x}\) in \({\mathbb R}^n\), then \({\mathbf y} = 0\).

   جواب رو نشون بده

   Hint:

   (a) \(\langle {\mathbf x}, {\mathbf y} \rangle = \langle {\mathbf y}, {\mathbf x} \rangle\).

6. Verify that \[\begin{aligned}\end{aligned}\] is a group.

7. Prove that \(\{ (2,1), (1,1) \}\) and \(\{ ( 12, 5), ( 7, 3) \}\) are bases for the same lattice.

   جواب رو نشون بده

   Hint:

   Use the unimodular matrix \[\begin{aligned}\end{aligned}\].

8. Let \(G\) be a subgroup of \(E(2)\) and suppose that \(T\) is the translation subgroup of \(G\). Prove that the point group of \(G\) is isomorphic to \(G/T\).

9. Let \(A \in SL_2({\mathbb R})\) and suppose that the vectors \({\mathbf x}\) and \({\mathbf y}\) form two sides of a parallelogram in \({\mathbb R}^2\). Prove that the area of this parallelogram is the same as the area of the parallelogram with sides \(A{\mathbf x}\) and \(A{\mathbf y}\).

10. Prove that \(SO(n)\) is a normal subgroup of \(O(n)\).

    جواب رو نشون بده

    Hint:

    Show that the kernel of the map \(\det : O(n) \rightarrow {\mathbb R}^*\) is \(SO(n)\).

11. Show that any isometry \(f\) in \({\mathbb R}^n\) is a one-to-one map.

12. Prove or disprove: an element in \(E(2)\) of the form \((A, {\mathbf x})\), where \({\mathbf x} \neq 0\), has infinite order.

13. Prove or disprove: There exists an infinite abelian subgroup of \(O(n)\).

    جواب رو نشون بده

    Hint:

    True.

14. Let \({\mathbf x} = (x_1, x_2)\) be a point on the unit circle in \({\mathbb R}^2\); that is, \(x_1^2 + x_2^2 = 1\). If \(A \in O(2)\), show that \(A {\mathbf x}\) is also a point on the unit circle.

15. Let \(G\) be a group with a subgroup \(H\) (not necessarily normal) and a normal subgroup \(N\). Then \(G\) is a semidirect product of \(N\) by \(H\) if

    • \(H \cap N = \{ \identity \}\);

    • \(HN=G\).

    Show that each of the following is true.

    1. \(S_3\) is the semidirect product of \(A_3\) by \(H = \{(1), (1 \,2) \}\).

    2. The quaternion group, \(Q_8\), cannot be written as a semidirect product.

    3. \(E(2)\) is the semidirect product of \(O(2)\) by \(H\), where \(H\) consists of all translations in \({\mathbb R}^2\).

16. Determine which of the \(17\) wallpaper groups preserves the symmetry of the pattern in .

17. Determine which of the \(17\) wallpaper groups preserves the symmetry of the pattern in .

    جواب رو نشون بده

    Hint:

    \(p6m\)

18. Find the rotation group of a dodecahedron.

19. For each of the \(17\) wallpaper groups, draw a wallpaper pattern having that group as a symmetry group.

Symbols used here

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

O(n^2),\ \Theta,\ \Omega

big-O notation
Grows no faster than n² (up to a constant), for large n.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

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حل

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Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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