Problems in elementary group theory

This is a problems page. The solutions can be found at the solutions page

This material will soon be incorporated in the guided tour for beginners. Go to the beginning of the guided tour for beginners for more details.

This page gives some problems in elementary group theory, meant as a resource for beginners to group theory to gauge their level of understanding. The problems are classified based on problem type, knowledge tested, and skills tested. Relevant material in the wiki for these problems can be found in:

Definition understanding

The problems in this section test understanding of definitions of group, subgroup, homomorphism of groups, normal subgroup, quotient group, automorphism of a group and trivial group.

True/False problems

If you have understood the definitions properly, you should be able to solve all these problems correctly. If you find any problem confusing, please review the relevant definitions. The solutions page contains, in addition tothe solutions, the relevant definitions for each problem.

1. The empty set can be given the structure of a group
2. For any group, there is a unique homomorphism to that group from the trivial group.
3. For any group, there is a unique homomorphism from that group to the trivial group.
4. If a subset of a group is nonempty and closed under multiplication, then that subset is a subgroup.
5. The identity element and inverse operation on a group are completely determined by the multiplication.
6. The binary operation on a group is completely determined by the identity element and inverse operation.
7. If $\varphi: G \to H$ is a homomorphism of groups, and $K$ is a subgroup of $H$, then $\varphi^{-1}(K)$ is a subgroup of $G$
8. If $\varphi: G \to H$ is a homomorphism of groups, and $K$ is a subgroup of $G$, then $\varphi(K)$ is a subgroup of $H$

Fact understanding

These problems are closely related to facts that may, for instance, be covered in Category:Elementary non-basic facts in group theory. They involve terms covered in Category:Basic definitions in group theory.

True/False problems

1. If a finite group can be expressed as the union of three proper subgroups, then at least one of them has index two.
2. If a finite group can be expressed as a union of four proper subgroups, then it can be expressed as a union of three proper subgroups.
3. If $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$, $H$ is a normal subgroup of $G$
4. If $H$ is a normal subgroup of $G$ and $\varphi:G \to K$ is a homomorphism, then $\varphi(H)$ is a normal subgroup of $K$
5. If $H$ and $K$ are subgroups of $G$ such that $H \cap K$ is the trivial subgroup and $HK = G$ (i.e. any element of $G$ can be expressed as $hk$ for $h \in H$ and $k \in K$, then $H$ and $K$ are normal
6. In the setup of the previous part, $KH = G$ as well
7. The union of all conjugates of a non-normal subgroup in a finite group, can never be a subgroup

A feel of some important group and subgroup properties

More problems on group properties can be found at Problems related to group properties, Problems related to subgroup properties

Classify as always true/true for Abelian/true for finite/other

Given below are some statements in the context of a group $G$. Figure out whether these statements are true for all $G$, for all finite groups $G$, for all Abelian groups $G$, or none of those. Give counterexamples if possible.

1. For any integer $n$, the set of elements $x^n : x \in G$, forms a subgroup
2. For any group $H$, the set of homomorphisms from $H$ to $G$ gets the structure of a group under pointwise multiplication
3. There exists an integer $n > 1$ such that the map $x \mapsto x^n$ is an endomorphism of $G$
4. For any integer $n$, the set of elements $x: x^n = e$ (here $e$ denotes the identity element) is a subgroup.
5. Every subgroup is normal
6. Suppose $H$ and $K$ are subgroups. Then the set of elements $hk, h \in H, k \in K$, denoted as the product of subgroups $HK$, is also a subgroup.

Normal and characteristic subgroups

Review, before starting, the definitions of normal subgroup and characteristic subgroup.

1. An intersection of a normal subgroup and a characteristic subgroup must be characteristic.
2. The trivial subgroup is characteristic.
3. If $H$ and $K$ are normal subgroups of $G$ such that $K$ is characteristic in $G$, then $K/H$ is characteristic in $G/H$.
4. If $H$ is a characteristic subgroup of $G$ and $K$ is an intermediate subgroup, then $H$ is a characteristic subgroup in $K$.
5. If a group $G$ is characteristic in the group $G \times G$ under the embedding $g \mapsto (g,e)$, then $G$ is trivial.
6. If $C$ is a union of conjugacy classes in a group $G$, the subgroup generated by $C$ is a normal subgroup.
7. A product of characteristic subgroups is again a subgroup, and is characteristic.
8. If every subgroup of a group is normal, then the group is Abelian.
9. For a cyclic group, every subgroup is characteristic.
10. If a subgroup of a finite group is the only subgroup of that order, then it is characteristic.
11. If a subgroup of a group permutes with every other subgroup (i.e. the product with any other subgroup is a subgroup) then the subgroup is normal.

Subgroup-defining functions

Refer the definitions of center, commutator subgroup, and Frattini subgroup, and read more about subgroup-defining function before starting this.

1. The center of any group is a characteristic subgroup.
2. For any group, the center and commutator subgroup intersect trivially.
3. Every subgroup of the group that lies inside the center, is characteristic.

Simple groups

Further information: simple group

1. An Abelian group is simple, if and only if it is cyclic of prime order
2. For a simple group, any endomorphism must be either injective or trivial
3. Given a simple normal subgroup $H$ in a group $G$, $H$ is a minimal normal subgroup: there is no nontrivial normal subgroup of $G$ contained properly inside $H$
4. Every group can be expressed as a direct product of simple groups
5. Every finite group can be embedded as a subgroup of a simple group

A feel of some particular groups

Cyclic and Abelian groups $p$ denotes a prime in all the exercises below,

1. In a cyclic group, it is true that given any two subgroups, one of them must be contained in the other
2. A direct product of cyclic groups is cyclic.
3. Any group is generated by its cyclic subgroups.
4. Among all Abelian groups of order $p^k$, the one with the minimum number of subgroups, is the cyclic one.
5. The number of isomorphism classes of Abelian groups of order $p^k$ is independent of $p$
6. In a non-Abelian group, the subgroup generated by two normal Abelian subgroups, is again a normal Abelian subgroup.
7. The number of Abelian subgroups of order $n$, is bounded from above by a polynomial in $n$
8. For any finite Abelian group, there exists an integer $d$, and an element $x$ such that $x$ has order exactly $d$, and any element of the group has order dividing $d$