Cosets and Lagrange's Theorem

Let $H = \{ 0, \pm 3, \pm 6, \pm 9, \ldots \}$. Find all the left cosets of $H$ in $\mathbb {Z}$. Let $n$ be a positive integer. Let $H = \{ 0, \pm n, \pm 2n, \pm 3n, \ldots \}$. Find all left cosets of $H$ in $\mathbb {Z}$. How many are there?

Rewrite the condition $a^{-1}b \in H$ given in property 6 of the lemma in this chapter in additive notation. Assume that the group is Abelian.

Let $H$ be as in Exercise 1. Use Exercise 2 to decide whether or not the following cosets of $H$ are the same.

Find all of the left cosets of $\{ 1, 11\}$ in $U(30)$.

Let $a$ belong to a group and $|a| = 30$. How many left cosets of $\left\langle a^5\right\rangle$ in $\left\langle a\right\rangle$ are there? List them. Do the same for $\left\langle a^4\right\rangle$ in $\left\langle a\right\rangle$.

Give an example of a group $G$ and subgroups $H$ and $K$ such that $HK = \left\{ hk \mid h \in H, k \in K\right\}$ is not a subgroup of $G$.

If $H$ is a subgroup of $\mathbb {Z}$ and $\left\langle 3\right\rangle \subseteq H \subseteq \mathbb {Z}$, prove that $H = \left\langle 3\right\rangle$ or $H = \mathbb {Z}$.

Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH = bK$, prove that $H = K$.

If $H$ and $K$ are subgroups of $G$ and $g$ belongs to $G$, show that $g(H \cap K) = gH \cap gK$.

Let $a$ and $b$ be nonidentity elements of different orders in a group $G$ of order 155. Prove that the only subgroup of $G$ that contains $a$ and $b$ is $G$ itself.

Let $H$ be a subgroup of $\mathbb {R}^*$, the group of nonzero real numbers under multiplication. If $\mathbb {R}^+ \subseteq H \subseteq \mathbb {R}^*$, prove that $H = \mathbb {R}^+$ or $H = \mathbb {R}^*$.

Let $\mathbb {C}^*$ be the group of nonzero complex numbers under multiplication and let $H = \left\{ a + bi \in \mathbb {C}^* \mid a^2 + b^2 = 1\right\}$. Give a geometric description of the coset $(3 + 4i)H$. Give a geometric description of the coset $(c + di)H$.

Let $G$ be a group of order 60. What are the possible orders for the subgroups of $G$?

Suppose that $K$ is a proper subgroup of $H$ and $H$ is a proper subgroup of $G$. If $|K| = 42$ and $|G| = 420$, what are the possible orders of $H$?

Let $G$ be a group with $|G| = pq$, where $p$ and $q$ are prime. Prove that every proper subgroup of $G$ is cyclic.

Recall that, for any integer $n$ greater than 1, $\phi (n)$ denotes the number of positive integers less than $n$ and relatively prime to $n$. Prove that if $a$ is any integer relatively prime to $n$, then $a^{\phi (n)} \bmod n = 1$.

Compute $5^{15} \bmod 7$ and $7^{13} \bmod 11$.

Use Corollary 2 of Lagrange's Theorem (Theorem 7.1) to prove that the order of $U(n)$ is even when $n > 2$.

(Corollary 2 of Theorem 7.1: In a finite group, the order of each element of the group divides the order of the group.)

Suppose $G$ is a finite group of order $n$ and $m$ is relatively prime to $n$. If $g \in G$ and $g^m = e$, prove that $g = e$.

Suppose $H$ and $K$ are subgroups of a group $G$. If $|H| = 12$ and $|K| = 35$, find $|H \cap K|$. Generalize.

For any integer $n \geq 3$, prove that $D_n$ has a subgroup of order 4 if and only if $n$ is even.

Let $H$ and $K$ be subgroups of an Abelian group. If $|H| = 12$ and $|K| = 18$ prove that $H \cap K$ is cyclic. Does your proof generalize to the case where $|H \cap K|$ divides $2p$ where $p$ is prime?

Suppose that $G$ is an Abelian group with an odd number of elements. Show that the product of all of the elements of $G$ is the identity.

Prove that every group of order 4 is Abelian.

Let $G$ be a group of order 25. Prove that $G$ is cyclic or $a^5 = e$ for all $a$ in $G$. Generalize to any group of order $p^n$ where $p$ is prime.

What is the maximum number of elements of order 11 in a group of order 99?

Let $|G| = 33$. What are the possible orders for the elements of $G$? Show that $G$ must have an element of order 3.

Suppose $G$ is a non-Abelian group with $|G| = 8$. Prove that $G$ has an element of order 4.

Prove that any group of order 55 must have exactly one subgroup of order 5 or exactly 11 subgroups of order 5.

Suppose $G$ is a group of order 105 with the property that $G$ has exactly one subgroup for each divisor of 105. Prove that $G$ is cyclic.

Use Exercise 87 of Chapter 4 to prove the following converse of the Fundamental Theorem of Finite Cyclic Groups: If $G$ is a finite group of order $n$ with the property that $G$ has exactly one subgroup of order $d$ for each positive divisor $d$ of $n$ then $G$ is cyclic.

(Exercise 87 of Chapter 4: for each positive integer $n$, $n = \sum \phi (d)$ where the sum ranges over all positive divisors $d$ of $n$.)

Suppose that $G$ is a group of order $2p$ where $p$ is an odd prime. What are the possibilities for the number of elements or order 2?

Let $G$ be a group with $|G| = 2m$ where $m$ is odd. Prove that $G$ has at most one subgroup of order $m$.

Determine all finite subgroups of $\mathbb {C}^*$, the group of nonzero complex numbers under multiplication.

Let $H$ and $K$ be subgroups of a finite group $G$ with $H \subseteq K \subseteq G$. Prove that $|G:H| = |G:K||K:H|$.

If a finite group $G$ has subgroups $H$ and $K$ such that $K \subseteq H \subseteq G$ with $[G:K] = p$ where $p$ is prime, prove that $H = G$ or $H = K$.

Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to $\mathbb {Z}_{12}$ and a subgroup isomorphic to $\mathbb {Z}_{20}$. No need to prove anything, but explain your reasoning.

Let $G$ be a group and $|G| = 21$. If $g \in G$ and $g^{14} = e$, what are the possibilities for $|g|$?

Suppose that a finite Abelian group $G$ has at least three elements of order 3. Prove that 9 divides $|G|$.

Prove that if $G$ is a finite group, the index of $Z(G)$ cannot be prime.

Let $p$ be a prime and $n$ be a positive integer. Prove that $a^{p^n} = a$ for every $a \in \mathbb {Z}_p$.

Prove that a group of order 63 must have an element of order 3.

Let $G$ be a group of order 100 that has a subgroup $H$ of order 25. Prove that every element of $G$ of order 5 is in $H$.

Let $G$ be a group of order $n$ and $k$ be any integer relatively prime to $n$. Show that the mapping from $G$ to $G$ given by $g \to g^k$ is one-to-one. If $G$ is also Abelian, show that the mapping given by $g \to g^k$ is an automorphism of $G$.

Let $G$ be a finite group and $d$ be the smallest positive integer such that $x^d = e$ for all $x$ in $G$. Prove that $d$ divides $|G|$.

Prove that every subgroup of $D_n$ of odd order is cyclic.

In a finite Abelian group prove that the number of elements of order 2 has the form $2^n - 1$.

Prove that an Abelian group of order 15 is cyclic.

Show that in a group $G$ of odd order, the equation $x^2 = a$ has a unique solution for all $a$ in $G$.

Prove that an Abelian group of order 20 has an element of order 5. Does your proof generalize to prove that every Abelian group of order $4p^n$ where $p$ is prime has a subgroup of order $p$?

Prove that a group of order $np^m$ where $p$ is prime and $n < p^m$ has at most one subgroup of order $p^m$.

Prove that a group $G$ of order $pq^n$ where $p$ and $q$ are primes and $p > q^n$ has an element of order $q$.

Suppose that a group of order 21 has exactly one subgroup of order 3. Prove that $G$ is cyclic. Does your proof work for groups of order $pq$ where $p < q$ are primes and there is a unique subgroups of order $q$?

Let $G$ be a finite group with more than one element and let $H$ be the intersection of all non-trivial subgroups of $G$. What can you say about the $|H|$?

Prove that a group that has more than one subgroup of order 5 must have order at least order 25. Generalize.

If $a$ is an element finite order in a group $G$ and $\phi_a$ is the inner automorphism of $G$ induced by $a$ (that is, $\phi_a(x) = axa^{-1}$ for all $x$ in $G$), prove that $|\phi_a|$ divides $|a|$.

If $G$ is a finite group with fewer than 100 elements and $G$ has subgroups of orders 10 and 25, what is the order of $G$?

Why does the fact that $A_4$ has no subgroup of order 6 imply that $|Z(A_4)| = 1$?

Use properties of cosets to prove that for any subgroup $H$ of $S_n$ either every member of $H$ is an even permutation or exactly half of the members is even. (See Exercise 27 in Chapter 5.)

Prove that $A_5$ has a subgroup of order 12.

Prove that $A_5$ has no subgroup of order 30.

Prove that $A_5$ has no subgroup of order 15 or 20.

Prove that $S_5$ has no subgroup of order 30.

Suppose that $\alpha \in S_n$ ($n \geq 3$) and $\alpha (1) = 3$. Let $H = \mathrm{stab}(1)$. Prove that $\alpha H$ is the set of all elements of $S_n$ that send 1 to 3. Generalize.

(Recall $\mathrm{stab}(a) = \left\{ \beta \in S_n \mid \beta (a) = a\right\}$, the stabilizer of $a$, from Exercise 43 of Chapter 5.)

Prove that $A_5$ is the only subgroup of $S_5$ of order 60.

For which positive integers $n$ is $S_n$ isomorphic to a subgroup of $D_{60}$. No need to prove anything but give a reason why any specific $n$ is or is not on your list.

Let $G$ be a group of permutations of a set $S$. Prove that the orbits of the members of $S$ constitute a partition of $S$. (This exercise is referred to in this chapter and in Chapter 27.)

The group $D_4$ acts as a group of permutations of the square regions shown below. (The axes of symmetry are drawn for reference purposes.) For each square region, locate the points in the orbit of the indicated point under $D_4$. In each case, determine the stabilizer of the indicated point.

(The figure shows six copies of a square, each with both diagonals and both midlines drawn as axes of symmetry, and a single marked point at a different position in each of the six copies: the midpoint of one side; a corner; a point on the horizontal midline that is off-center (not at the center); a point on a side that is off-center and not at a corner; and, in the remaining two copies, a point that lies off every axis of symmetry, in the region between the vertical midline and a diagonal.)

Let $G = \{ (1), (12)(34), (1234)(56), (13)(24), (1432)(56), (56)(13), (14)(23), (24)(56)\}$.

Noting that the rotation group of a cube is isomorphic to $S_4$ and that $S_4$ has 8 elements of order 3, we know that there are 8 rotations of 120 degrees that are symmetries of the cube. Give a geometrical description of those 8 rotations.

Let $G = GL(2, \mathbb {R})$ and $H = SL(2, \mathbb {R})$. Let $A \in G$ and suppose that $\det A = 2$. Prove that $AH$ is the set of all $2 \times 2$ matrices in $G$ that have determinant 2.

Let $G$ be the group of rotations of a plane about a point $P$ in the plane. Thinking of $G$ as a group of permutations of the plane, describe the orbit of a point $Q$ in the plane. (This is the motivation for the name ``orbit.'')

Let $G$ be the rotation group of a cube. Label the faces of the cube 1 through 6, and let $H$ be the subgroup of elements of $G$ that carry face 1 to itself. If $\sigma$ is a rotation that carries face 2 to face 1, give a physical description of the coset $H\sigma$.

A soccer ball has 20 faces that are regular hexagons and 12 faces that are regular pentagons. Use Theorem 7.4 to explain why a soccer ball cannot have a $60^\circ$ rotational symmetry about a line through the centers of two opposite hexagonal faces.

(Theorem 7.4, the Orbit-Stabilizer Theorem: Let $G$ be a finite group of permutations of a set $S$. Then, for any $i$ from $S$, $|G| = |\mathrm{orb}_G(i)||\mathrm{stab}_G(i)|$.)

Let $G = GL(2, \mathbb {R})$, the group of $2 \times 2$ matrices over $\mathbb {R}$ with nonzero determinant. Let $H$ be the subgroup of matrices of determinant $\pm 1$. If $a, b \in G$ and $aH = bH$, what can be said about $\det (a)$ and $\det (b)$? Prove or disprove the converse.

Calculate the orders of the following (refer to Figure 26.5 for illustrations).

Prove that the eight-element set in the proof of Theorem 7.5 is a group.

(Theorem 7.5, The Rotation Group of a Cube, is proved using two rotations $\alpha = (1234)$ and $\beta = (1423)$ of the four diagonals of a cube; the eight-element set in question is $\{ \varepsilon , \alpha , \alpha^2, \alpha^3, \beta^2, \beta^2\alpha , \beta^2\alpha^2, \beta^2\alpha^3\}$.)