Algebra

Let $G$ be the collection of $2\times 2$ real matrices with nonzero determinant. Define the product of two elements in $G$ as the usual matrix product.

Let $G$ be the set of $3 \times 3$ real matrices with zeros below the diagonal and ones on the diagonal.

For each of the following either give an example or else prove that no such example is possible.

Let $R$ be a ring with multiplicative identity 1. Call $x \in R$ a unit if $xy = yx = 1$ for some $y \in R$. Let $G(R)$ denote the set of units.

In the triangular network in $\mathbb {R}^{2}$ depicted below, the points $P_{0}, P_{1}, P_{2}$, and $P_{3}$ are respectively $(0,0), (1,0), (0,1)$, and $(1,1)$. Describe the structure of the group of all Euclidean transformations of $\mathbb {R}^{2}$ which leave this network invariant.

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Does the set $G = \{ a \in \mathbb {R} \mid a > 0, a \neq 1\}$ form a group with the operation $a * b = a^{\log b}$?

Let $G$ be a finite group. A conjugacy class is a set of the form

for some $a \in G$.

Let $G$ be a finite nontrivial group with the property that for any two elements $a$ and $b$ in $G$ different from the identity, there is an element $c$ in $G$ such that $b = c^{-1}ac$. Prove that $G$ has order 2.

Let $G$ be a finite group, with identity $e$. Suppose that for every $a, b \in G$ distinct from $e$, there is an automorphism $\sigma$ of $G$ such that $\sigma (a) = b$. Prove that $G$ is abelian.

For a $p$-group of order $p^{4}$, assume the center of $G$ has order $p^{2}$. Determine the number of conjugacy classes of $G$.

In a commutative group $G$, let the element $a$ have order $r$, let $b$ have order $s$ ($r, s < \infty$), and assume that the greatest common divisor of $r$ and $s$ is 1. Show that $ab$ has order $rs$.

Let $G$ be a group. For any subset $X$ of $G$, define its centralizer $C(X)$ to be $\{ y \in G \mid xy = yx, \text{ for all } x \in X\}$. Prove the following:

Let $D$ be a group of order $2n$, where $n$ is odd, with a subgroup $H$ of order $n$ satisfying $xhx^{-1} = h^{-1}$ for all $h$ in $H$ and all $x$ in $D \setminus H$. Prove that $H$ is commutative and that every element of $D \setminus H$ is of order 2.

How many homomorphisms are there from the group $\mathbb {Z}_{2} \times \mathbb {Z}_{2}$ to the symmetric group on three objects?

The set of $5 \times 5$ complex matrices $A$ satisfying $A^{3} = A^{2}$ is a union of conjugacy classes. How many conjugacy classes?

Let $\mathbb {C}^{*}$ be the multiplicative group of nonzero complex numbers. Suppose that $H$ is a subgroup of finite index of $\mathbb {C}^{*}$. Prove that $H = \mathbb {C}^{*}$.

Let $G$ be a finite group and $H \subset G$ a subgroup.

Prove that the group of automorphisms of a cyclic group of prime order $p$ is cyclic and find its order.

Let $G$ be a finite group, and let $\varphi$ be an automorphism of $G$ which leaves fixed only the identity element of $G$.

Prove that every finite group of order $> 2$ has a nontrivial automorphism.

Let $A$ be an additively written abelian group, and $u, v : A \to A$ homomorphisms. Define the group homomorphisms $f, g : A \to A$ by

Prove that the kernel of $f$ is isomorphic to the kernel of $g$.

Let $G$ be an additive group, and $u, v : G \to G$ homomorphisms. Show that the map $f : G \to G$, $f(x) = x - v(u(x))$ is surjective if the map $h : G \to G$, $h(x) = x - u(v(x))$ is surjective.

Let $H$ be the group of integers mod $p$, under addition, where $p$ is a prime number. Suppose that $n$ is an integer satisfying $1 \leq n \leq p$, and let $G$ be the group $H \times H \times \cdots \times H$ ($n$ factors). Show that $G$ has no automorphism of order $p^{2}$.

Let $G$ be a group and $H$ a subgroup of index $n < \infty$. Prove or disprove the following statements:

Find all automorphisms of the additive group of rational numbers.

Let $\alpha$ and $\beta$ be real numbers such that the subgroup $\Gamma$ of $\mathbb {R}$ generated by $\alpha$ and $\beta$ is closed. Prove that $\alpha$ and $\beta$ are linearly dependent over $\mathbb {Q}$.

Prove that the group $G = \mathbb {Q}/\mathbb {Z}$ has no proper subgroup of finite index.

Let $A$ be the group of rational numbers under addition, and let $M$ be the group of positive rational numbers under multiplication. Determine all homomorphisms $\varphi : A \to M$.

Suppose that $H_{1}$ and $H_{2}$ are distinct subgroups of a group $G$ such that $[G:H_{1}] = [G:H_{2}] = 3$. What are the possible values of $[G:H_{1} \cap H_{2}]$?

Let $G$ be a group and $H$ and $K$ subgroups such that $H$ has a finite index in $G$. Prove that $K \cap H$ has a finite index in $K$.

Suppose the group $G$ has a nontrivial subgroup $H$ which is contained in every nontrivial subgroup of $G$. Prove that $H$ is contained in the center of $G$.

Show that for each positive integer $k$ there exists a positive integer $N$ such that there are at least $k$ nonisomorphic groups of order $N$.

Let $G$ be a group generated by $n$ elements. Find an upper bound $N(n,k)$ for the number of subgroups $H$ of $G$ with the index $[G:H] = k$.

Let $G$ be the group $\mathbb {Q}/\mathbb {Z}$. Show that for every positive integer $t$, $G$ has a unique cyclic subgroup of order $t$.

Show that a group $G$ is isomorphic to a subgroup of the additive group of the rationals if and only if $G$ is countable and every finite subset of $G$ is contained in an infinite cyclic subgroup of $G$.

Prove that any group of order 77 is cyclic.

Let $G$ be a group of order $2p$, where $p$ is an odd prime. Assume that $G$ has a normal subgroup of order 2. Prove that $G$ is cyclic.

A finite abelian group $G$ has the property that for each positive integer $n$ the set $\{ x \in G \mid x^{n} = 1\}$ has at most $n$ elements. Prove that $G$ is cyclic, and deduce that every finite field has cyclic multiplicative group.

Let $G$ be a finite group of order $n$ with the property that for each divisor $d$ of $n$ there is at most one subgroup in $G$ of order $d$. Show $G$ is cyclic.

Let $H$ be a subgroup of a finite group $G$.

Let $G$ be the group of orthogonal transformations of $\mathbb {R}^{3}$ to $\mathbb {R}^{3}$ with determinant 1. Let $v \in \mathbb {R}^{3}$, $\left|v\right| = 1$, and let $H_{v} = \{ T \in G \mid Tv = v\}$.

Show that if a subgroup $H$ of a group $G$ has just one left coset different from itself then it is a normal subgroup of $G$.

Let $G$ be a group of order 120, let $H$ be a subgroup of order 24, and assume that there is at least one left coset of $H$ (other than $H$ itself) which is equal to some right coset of $H$. Prove that $H$ is a normal subgroup of $G$.

Let $G$ be a group of order 112. Prove that $G$ has a nontrivial normal subgroup.

List all groups of order 6 up to isomorphism. Find a group of order 120 that contains as a subgroup an isomorphic copy of each of them. Prove that no group of order $< 120$ has the preceding property.

For $G$ a group and $H$ a subgroup, let $C(G,H)$ denote the collection of left cosets of $H$ in $G$. Prove that if $H$ and $K$ are two subgroups of $G$ of infinite index then $G$ is not a finite union of cosets from $C(G,H) \cup C(G,K)$.

Let

Let $\mathbb {Z}^{2}$ be the group of lattice points in the plane (ordered pairs of integers, with coordinatewise addition as the group operation). Let $H_{1}$ be the subgroup generated by the two elements $(1,2)$ and $(4,1)$, and $H_{2}$ the subgroup generated by the two elements $(3,2)$ and $(1,3)$. Are the quotient groups $G_{1} = \mathbb {Z}^{2}/H_{1}$ and $G_{2} = \mathbb {Z}^{2}/H_{2}$ isomorphic?

Let $G$ be a group of order 10 which has a normal subgroup of order 2. Prove that $G$ is abelian.

Let $G$ be a group with three normal subgroups $N_{1}$, $N_{2}$, and $N_{3}$. Suppose $N_{i} \cap N_{j} = \{ e\}$ and $N_{i}N_{j} = G$ for all $i, j$ with $i \neq j$. Show that $G$ is abelian and $N_{i}$ is isomorphic to $N_{j}$ for all $i, j$.

Suppose $H_{i}$ is a normal subgroup of a group $G$ for $1 \leq i \leq k$, such that $H_{i} \cap H_{j} = \{ 1\}$ for $i \neq j$. Prove that $G$ contains a subgroup isomorphic to $H_{1} \times H_{2} \times \cdots \times H_{k}$ if $k = 2$, but not necessarily if $k \geq 3$.

$G$ is a group of order $n$, $H$ a proper subgroup of order $m$, and $(n/m)! < 2n$. Prove $G$ has a proper normal subgroup different from the identity.

Prove that if $G$ is a group containing no subgroup of index 2, then any subgroup of index 3 is normal.

Suppose $G$ is a nonabelian simple group, and $A$ is its automorphism group. Show that $A$ contains a normal subgroup isomorphic to $G$.

Let $G$ be a group whose order is twice an odd number. For $g$ in $G$, let $\lambda_{g}$ denote the permutation of $G$ given by $\lambda_{g}(x) = gx$ for $x \in G$.

Let $G$ be a group, $G'$ its commutator subgroup, and $N$ a normal subgroup of $G$. Suppose that $N$ is cyclic. Prove that $gn = ng$ for all $g \in G'$ and all $n \in N$.

Let $G$ be a group and $N$ be a normal subgroup of $G$ with $N \neq G$. Suppose that there does not exist a subgroup $H$ of $G$ satisfying $N \subset H \subset G$ and $N \neq H \neq G$. Prove that the index of $N$ in $G$ is finite and equal to a prime number.

Let $G$ be a group having a subgroup $A$ of finite index. Prove that there is a normal subgroup $N$ of $G$ contained in $A$ such that $N$ is of finite index in $G$.

Let $H$ be the quotient of an abelian group $G$ by a subgroup $K$. Prove or disprove each of the following statements:

Let $\mathbb {F}_{2} = \{ 0,1\}$ be the field with two elements. Let $G$ be the group of invertible $2 \times 2$ matrices with entries in $\mathbb {F}_{2}$. Show that $G$ is isomorphic to $S_{3}$, the group of permutations of three objects.

Let $S_{n}$ denote the group of permutations of $n$ objects. Find four different subgroups of $S_{4}$ isomorphic to $S_{3}$ and nine isomorphic to $S_{2}$.

Let $G$ be a subgroup of $S_{5}$, the group of all permutations of five objects. Prove that if $G$ contains a 5-cycle and a 2-cycle, then $G = S_{5}$.

Let $G$ be a subgroup of the symmetric group on six objects, $S_{6}$. Assume that $G$ has an element of order 6. Prove that $G$ has a normal subgroup $H$ of index 2.

Let $S_{7}$ be the group of permutations of a set of seven objects. Find all $n$ such that some element of $S_{7}$ has order $n$.

$S_{9}$ is the group of permutations of 9 objects.

Let $S_{9}$ denote the group of permutations of nine objects and let $A_{9}$ be the subgroup consisting of all even permutations. Denote by $1 \in S_{9}$ the identity permutation. Determine the minimum of all positive integers $m$ such that every $\sigma \in S_{9}$ satisfies $\sigma^{m} = 1$. Determine also the minimum of all positive integers $m$ such that every $\sigma \in A_{9}$ satisfies $\sigma^{m} = 1$.

Let $S_{999}$ denote the group of permutations of 999 objects, and let $G \subset S_{999}$ be an abelian subgroup of order 1111. Prove that there exists $i \in \{ 1, \ldots , 999\}$ such that for all $\sigma \in G$, one has $\sigma (i) = i$.

Prove that $S_{n}$, the group of permutations of $\{ 1,2,\ldots ,n\}$, is isomorphic to a subgroup of $A_{n+2}$, the alternating subgroup of $S_{n+2}$.

Let $S_{n}$ be the group of all permutations of $n$ objects and let $G$ be a subgroup of $S_{n}$ of order $p^{k}$, where $p$ is a prime not dividing $n$. Show that $G$ has a fixed point; that is, one of the objects is left fixed by every element of $G$.

Let $G$ be a subgroup of $S_{n}$, the group of permutations of $n$ objects. Assume $G$ is transitive; that is, for any $x$ and $y$ in $S$, there is some $\sigma \in G$ with $\sigma (x) = y$.

Let $G$ be a transitive subgroup of the group $S_{n}$ of permutations of $n$ objects $\{ 1, \ldots , n\}$. Suppose that $G$ is a simple group and that $\sim$ is an equivalence relation on $\{ 1, \ldots , n\}$ such that $i \sim j$ implies that $\sigma (i) \sim \sigma (j)$ for all $\sigma \in G$. What can one conclude about the relation $\sim$?

Let $D_{n}$ be the dihedral group, the group of rigid motions of a regular $n$-gon ($n \geq 3$). (It is a noncommutative group of order $2n$.) Determine its center $Z = \{ c \in D_{n} \mid cx = xc \text{ for all } x \in D_{n}\}$.

How many Sylow 2-subgroups does the dihedral group $D_{n}$ of order $2n$ have, when $n$ is odd?

Let $G$ be a finite group and suppose that $G \times G$ has exactly four normal subgroups. Show that $G$ is simple and nonabelian.

Let $G$ be a finite simple group of order $n$. Determine the number of normal subgroups in the direct product $G \times G$.

Prove that $\mathbb {Q}$, the additive group of rational numbers, cannot be written as the direct sum of two nontrivial subgroups.

Note: See also Problems 6.3.1 and 6.7.2.

Let $A$, $B$, and $C$ be finite abelian groups such that $A \times B$ and $A \times C$ are isomorphic. Prove that $B$ and $C$ are isomorphic.

Let $G_{1}$, $G_{2}$, and $G_{3}$ be finite groups, each of which is generated by its commutators (elements of the form $xyx^{-1}y^{-1}$). Let $A$ be a subgroup of $G_{1} \times G_{2} \times G_{3}$, which maps surjectively, by the natural projection map, to the partial products $G_{1} \times G_{2}$, $G_{1} \times G_{3}$ and $G_{2} \times G_{3}$. Show that $A$ is equal to $G_{1} \times G_{2} \times G_{3}$.

Let $A$ be a subgroup of an abelian group $B$. Assume that $A$ is a direct summand of $B$, i.e., there exists a subgroup $X$ of $B$ such that $A \cap X = 0$ and such that $B = X+A$. Suppose that $C$ is a subgroup of $B$ and satisfying $A \subset C \subset B$. Is $A$ necessarily a direct summand of $C$?

Let $G$ and $H$ be finite groups of relatively prime order. Show that $\operatorname {Aut}(G \times H)$, the group of automorphisms of $G \times H$, is isomorphic to the direct product of $\operatorname {Aut}(G)$ and $\operatorname {Aut}(H)$.

Let $p$ be a prime, and let $G$ be the group $\mathbb {Z}_{p^{2}} \times \mathbb {Z}_{p}$. How many automorphisms does $G$ have?

Let $\mathbb {Q}_{+}$ be the multiplicative group of positive rational numbers.

Prove that the additive group of $\mathbb {Q}$, the rational number field, is not finitely generated.

Note: See also Problems 6.3.1 and 6.6.3.

Let $G$ be the abelian group defined by generators $x$, $y$, and $z$, and relations

Suppose that the group $G$ is generated by elements $x$ and $y$ that satisfy $x^{5}y^{3} = x^{8}y^{5} = 1$. Is $G$ the trivial group?

Let $G$ be a group with generators $a$ and $b$ satisfying

Is $G$ trivial?

Let the group $G$ be generated by two elements, $a$ and $b$, both of order 2. Prove that $G$ has a subgroup of index 2.

Let $G_{n}$ be the free group on $n$ generators. Show that $G_{2}$ and $G_{3}$ are not isomorphic.

Let $G$ be an abelian group which is generated by, at most, $n$ elements. Show that each subgroup of $G$ is again generated by, at most, $n$ elements.

Determine all finitely generated abelian groups $G$ which have only finitely many automorphisms.

Let $A$ be a finite abelian group, and $m$ the maximum of the orders of the elements of $A$. Put $S = \{ a \in A \mid \left|a\right| = m\}$. Prove that $A$ is generated by $S$.

List, to within isomorphism, all the finite groups whose orders do not exceed 5.

Show that all groups of order $\leq 5$ are commutative. Give an example of a noncommutative group of order 6.

Prove that any group of order 6 is isomorphic to either $\mathbb {Z}^{6}$ or $S_3$ (the group of permutations of three objects).

Find all abelian groups of order 8, up to isomorphism. Then identify which type occurs in each of

Show that there are at least two nonisomorphic nonabelian groups of order 24, of order 30 and order 40.

List eight groups of order 36 and prove that they are not isomorphic.

Prove that if $p$ is prime then every group of order $p^2$ is abelian.

Classify up to isomorphism all groups of order 45.

Let $G$ be a group of order 56 having at least 7 elements of order 7. Prove that $G$ has only one Sylow 2-subgroup $P$, and that all nonidentity elements of $P$ have order 2.

Classify all abelian groups of order 80 up to isomorphism.

Find (up to isomorphism) all groups of order $2p$, where $p$ is a prime ($p \geq 2$).