Group Explorer 3.0 Help: Group Theory Terminology

This page is a dictionary of many of the essential terms one might come across when beginning to learn group theory. Many other Group Explorer help pages link here to define terms. Unlike the Group Explorer terminology page, these terms not specific to Group Explorer itself; all are all commonly used mathematical terms.

1-1 (“one-to-one”)

See injective.

Abelian group

An abelian group is one whose binary operation is commutative. That is, for every two elements $a$ and $b$ in the group, $a\cdot b=b\cdot a$.

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Bijection, bijective

A function that is both injective and surjective is called bijective.

Cayley diagrams

A Cayley diagram is a graph (that is, a set of vertices and edges among them) that depicts a group. There is one node (vertex) in the graph for each element in the group, and the arrows (edges) show how the generators act on the elements of the group.

For instance, if the group has two generators, $a$ and $b$, then there will be one type of arrow (perhaps red-colored arrows) for generator $a$ and another type of arrow (perhaps blue-colored) for generator $b$. You can see that the following Cayley diagram fits this description.

The red arrows connect two elements if multiplying the first by $a$ gives the second. That is, we have $x~{\color{red}\longrightarrow}~y$ just when $x\cdot a=y$. So the arrows represent right-multiplication. (One could also make Cayley diagrams in which the arrows represented left-multiplication.)

Group Explorer has a visualizer for showing Cayley diagrams. It is documented in full here, with an introduction here, and a tutorial here.

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Cayley table

See multiplication table.

Class equation

The elements of a group can be partitioned into conjugacy classes. The class equation is a numerical equation describing this partitioning. For instance, the group $S_3$ has three conjugacy classes, of sizes 1, 2, and 3 respectively. The order of the group is 6, and so the class equation is $$1 + 2 + 3 = 6.$$

All class equations are of this form: The left hand side is a sum of positive integers, each the size of a conjugacy class, and the right side the order of the group.

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Closure of a subset

Not all subsets of a group are subgroups. The closure of a subset is the smallest subgroup containing that subset. That is, it answers the question, “What must I add to this subset to get a subgroup?” Another way to think of a subset’s closure is that it is the subgroup for which the subset is a set of generators.

Commutative group

See Abelian.

Conjugacy classes

The conjugacy class of an element $g$ in a group is the set of all elements $hgh^{-1}$ for any element $h$ in the group. These are called “classes” because they partition the group (that is, they form an equivalence relation) and are called “conjugacy” classes because the element $hgh^{-1}$ is called the “conjugate” of $g$ by $h$.

See also class equation.

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Cosets

For any subgroup $H$ of a group $G$, we can speak of the left cosets and the right cosets of $H$. The notation $aH$ for an element $a$ in $G$ means the set of all products $a$ times an element of $H$; this set is a left coset of $H$, because we multiplied by $a$ on the left of $H$. One could do the same on the right and form a right coset of $H$. The collection of all $aH$ for every $a$ in $G$ is the collection of left cosets of $H$; similarly for right cosets. Cosets of either type partition the elements of the group.

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Cycle graph

A cycle graph is an illustration of the cycles of a group (orbits of elements) and how those cycles connect. Here is an example cycle graph.

The above graph shows the group $\mathbb{Z}_2\times\mathbb{Z}_4$. One can see that there are two four-cycles (in the top half of the picture) which share two nodes (the central node and the topmost node). In addition to these six elements, there are two other order-2 elements that are not in either of the two larger four-cycles, shown at the bottom of the picture.

Group Explorer has a visualizer for showing you Cycle graphs, documented in full here.

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Cyclic group

A cyclic group is one that is generated by one element. Therefore it is comprised entirely of the orbit of that element. The Cyclic groups are denoted $\mathbb{Z}_n$ or sometimes $C_n$, and look just like their name (cycles) when viewed as Cayley diagrams or cycle graphs.

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Definition of a group via generators and relations

One can define groups by listing generators $a,b,c,\ldots$ and then writing equations that describe how they relate. For instance, the cyclic group with five elements can be described as the group generated by $a$ where $a^5$ is the identity element. This is written $$\left\langle a : a^5=e\right\rangle.$$ The portion to the left of the colon (:) is the lone generator $a$ and the portion to the right is the equation that describes it.

For groups with more than one generator, the situation is slightly more complex, but is essentially the same. The definition $$\left\langle a,b : a^4=e, b^2=e, ab=ba\right\rangle$$ describes a group with an order-4 generator $a$ and an order-2 generator $b$ which commute with each other. The last equation $ab=ba$ describes the commutativity of the generators, and therefore implies the commutativity of the whole group. This group is thus $\mathbb{Z}_2\times\mathbb{Z}_4$. But if we had written the same definition with a different final equation, say $bab=a^{-1}$, we would have come up with a different group ($D_4$).

One of the columns available for view in the main page (the group library) is the definition in this format of every group loaded.

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Elementwise product (of two subsets of a group)

The elementwise product of two subsets $S$ and $T$ of a group is written $ST$ and is the set of all products of elements from $S$ with elements from $T$, in that order. Thus the product is taken on the level of elements, or “elementwise.” In set theory notation, $$ST=\left\{ st : s\text{ is from }S\text{ and }t\text{ is from }T \right\}.$$

Epimorphism

See surjection.

First Isomorphism Theorem

The First Isomorphism Theorem says that given a homomorphism $f$ mapping a group $G$ to a group $G'$, the kernel of $f$ is a normal subgroup of $G$, and when we take the quotient $\frac{G}{\text{Ker }f}$, it is isomorphic to the image $\text{Im}(f)$ as a subgroup of $G'$.

This is useful in short exact sequences, which Group Explorer uses to illustrate the normality of subgroups. (To see an example, open a group info page, click the “tell me more” link next to the Subgroups computation, and search for “short exact sequence” on that page–it will be in the description of any normal subgroup.)

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Generators for a group (or subgroup)

A collection $C$ of elements of a group is said to generate the group (and they are called generators) if all possible combinations of multiplications of those elements with one another yields all elements of the group.

For instance, the elements $r$ and $f$ in $S_3$ generate the group because the complete list of elements of $S_3$ is $r,f,rf,fr,r^2$, plus the identity, which can be written as $r^3$. Thus all elements of $S_3$ are expressible as products of $r$s and $f$s, so the set $\{r,f\}$ generates $S_3$. Sometimes this is written $\left\langle r,f \right\rangle=$ $S_3$.

If we consider just the element $r$ in the same group and ask what set of elements it generates, $\langle r\rangle$, we find only the elements $\{e,r,r^2\}$. Therefore $r$ does not generate all of $S_3$, but only a subgroup.

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Group

A group is a set of elements together with a binary operation (which I’ll denote here by $*$) such that the following criteria hold.

• The binary operation is associative: For every $a$, $b$, and $c$ in the set, $a*(b*c)=(a*b)*c$.
• There is a special element in the set called the identity, which I will here denote $e$. The identity obeys this rule: for every $a$ in the group, $a*e=e*a=a$.
• For every element $a$ in the set, there is an element that is its inverse, usually written $a^{-1}$, such that $a*a^{-1} = e$.

But this is the formal definition of a group. You should be looking at pictures of them! That’s what this program is for!

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Homomorphism

Sometimes called simply a morphism, this is a function from one group to another–that is, from the set of elements of the one group to the set of elements of the other–that also preserves the group operation. That is, it must keep the group’s structure intact in the following specific way: For every pair of elements $a$ and $b$ in the domain, the homomorphism $f$ must satisfy the equation $f(ab) = f(a)f(b)$. “Homomorphism” means “same shape/form” (homo = same, morph = shape/form).

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Image (of a subset under a morphism)

If a homomorphism $f$ maps group $G$ to group $G'$, and there is a subset $S$ of $G$, we can find out what its “image” is under the homomorphism $f$ by simply applying $f$ to each element of $S$. That is, the image of $S$ under $f$, sometimes written $f[S]$, is the set $\left\{ f(a) : a\text{ is in }S \right\}$. Obviously this will be a subset of $G'$.

One can also speak of the image of $f$, meaning the image of the whole group $G$ under $f$, i.e. all elements of the codomain of $f$.

See also preimage.

Index of a subgroup

The order of a subgroup will always divide the order of the group. So we say that a subgroup’s index is the integer resulting from that division. In symbols, if $H$ is a subgroup of $G$, then because $|H|$ divides $|G|$, the index is $\frac{|G|}{|H|}$, often written $[G:H]$.

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Injective, injection

A function is injective (or 1-1, or an injection) if no two different elements map to the same output. That is, if $a$ and $b$ are not equal, then $f(a)$ and $f(b)$ are not equal either. An injective homomorphism is sometimes called a monomorphism.

Isomorphism, isomorphic

A homomorphism that is bijective is called an isomorphism. In group theory, if there is an isomorphism from one group to another, that means that those groups are really the same exact structure, but possibly with different names or labels. In other words, they are the same mathematical object.

Oftentimes people do not consider the distinction of labeling significant, and will therefore call two objects the same if they are isomorphic. This is what is meant by the phrase “the same up to isomorphism”; it means the objects are the same if we consider isomorphic things to be identical.

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The isomorphism theorems

The First Isomorphism Theorem can be illustrated in Group Explorer. Other isomorphism theorems do not yet appear illustrated in this software.

Kernel (of a homomorphism)

The kernel of a homomorphism is the set of elements that the homomorphism maps to the identity element. Sometimes written $\text{Ker}(f)$, this set is the preimage of the set $\{ e \}$ (where $e$ is the identity of the group to which the homomorphism maps), in symbols $\{ a : f(a) = e \}$.

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Lattice of subgroups

I will not define the concept of a lattice here; it is too broad a subject for the small need we have of it. One can find a good definition of it in a discrete mathematics textbook or online.

For our purposes, a lattice is a two-dimensional arrangement of sets, with larger objects higher in the arrangement (vertically), and with arrows drawn from smaller objects up to larger ones if the smaller object is a subset of the larger. Because all subgroups of a group are sets, we can arrange them in a lattice. For example, click here to see the lattice of subgroups of $S_3$ (illustrated using multiplication tables).

Left cosets

See cosets.

Monomorphism

See injection.

Morphism

See homomorphism.

Multiplication table

A multiplication table for a group is so named because it is much like elementary school multiplication tables, except that it uses the group elements and operation rather than integers under ordinary, everyday multiplication.

Thus the table is a grid, and across the top row and down the left column every element of the group is listed, and filling the rest of the table are the results of applying the group operation to the elements in the header row and column. For this reason, multiplication tables very well exhibit patterns inherent in the group operation, but elements themselves appear several times in the table, and thus the group as a set is not well depicted.

Group Explorer has a visualizer for showing you multiplication tables. More information about it appears in its documentation, introduction, and tutorial.

Group Explorer shows multiplication tables in one of two ways–with or without text in the cells of the table. Consider the following multiplication table for the group $D_4$.

This table has no text and thus the colors of the cells exhibit the abstract pattern inherent in the group operation. Omitting the text allows tables to be shown in small sizes (like in group info pages or the group library). If you want to see this same table with the element names in the cells, click here to open the group info page for $D_4$ and then click the multiplication table visualizer shown under the Views section. (Or jump to it directly with this link.)

Upon inspecting the multiplication table with text included, you can see that for example the element $r$ in the leftmost column, second row, multiplied by the element $f$ (in the topmost row, fifth column) results in the element $rf$, in the second row, fifth column. This is because multiplication is done in the order you can infer from this example: element in left column times element in top row.

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Normal subgroup

A subgroup is called normal when any one of the following equivalent criteria are met.

• A normal subgroup is one whose collection of left cosets is the same as its collection of right cosets.
• A normal subgroup is one which is self-conjugate, that is if $N$ is the subgroup then for any element $G$ in the group, the set $gNg^{-1}=N$. Here $gNg^{-1}=\{ gng^{-1} : n\text{ is in }N\}$.
• A normal subgroup $N$ in a group $G$ is one for which the quotient $\frac{G}{N}$ is defined.

The last of these is probably the easiest to visualize. Multiplication tables and Cayley diagrams can both organize themselves by the cosets of a subgroup and then separate those cosets (or “chunk” them) to help you visualize the quotient operation. Refer to the documentation on the multiplication table interface or the Cayley diagram interface for more information on these features.

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Normalizer of a subgroup/subset

The normalizer of a subgroup $H$ of a group $G$, sometimes written $\text{Norm}(H)$, is the largest subgroup containing $H$ in which $H$ is normal.

That is, $H$ may not be normal in $G$, but if we were to remove some of the “problem” elements from $G$, those that are preventing $H$ from being normal, we would find a subgroup of $G$ in which $H$ is normal. The normalizer is exactly this, the subgroup which remains when you remove as few elements as possible from $G$ to make $H$ normal.

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Objects of symmetry

Group theory is the study of symmetry in the abstract. But many very concrete objects which one could hold in one’s hand have symmetry (or symmetries) to them, and the relationship among those symmetries can be described by a group. Therefore several groups in Group Explorer’s library are groups that describe the symmetry of objects that exist in three-dimensional space, and exist in the real world–for example, a cube, a pinwheel, or a pyramid. These objects are called “objects of symmetry” (or objects with symmetry, or symmetry objects).

Group Explorer has a visualizer for showing you objects of symmetry, documented in full here.

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One-to-one

See injective.

Onto

See surjective.

Orbit of an element in a group

The orbit of an element a in a group $G$ is the set of all powers of that element, i.e. $\{ a^n : n\text{ is any integer} \}$. You can think of this as all the places one can “walk” using $a$ as a step. In a Cayley diagram, this would be the chain of elements one encounters by following $a$-arrows repeatedly.

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Order classes

Each element in a group has an order, and thus we can partition the elements of the group into classes which all have the same order. For instance the elements of the group $S_3$ are listed in the table below, with their orders.

Thus this group has three order classes: one consisting of the elements of order 1, $\{ e \}$, another consisting of the elements of order 2, $\{ f, fr, rf \}$, and another consisting of the elements of order 3, $\{ r, r^2 \}$.

You can learn about the order classes of any group by looking under the Computations section of its group info page.

Order of an element in a group

The order of an element in a group can be thought of in two equivalent ways.

• The order of the element $a$ is the smallest positive power of a that yields the identity. That is, if $a^k=e$ and no smaller positive integer exponent satisfies that same equation, then the order of $a$, written $|a|$, is $k$.
• The order of the element $a$ is the order of the subgroup generated by $a$, which is also the orbit of $a$.

The order of the identity element is 1 by either of these reckonings. It generates the subgroup $\{ e \}$.

Order of a group

The order of a group $G$, written $|G|$, is simply its size (how many elements are in it).

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Order of a subgroup

The order of a subgroup $H$, written $|H|$, is simply its size (how many elements are in it). See also subgroup index.

$p$-subgroup

A $p$-subgroup is a subgroup all of whose elements have an order equal to a power of the prime number $p$. (See also Sylow $p$-subgroup.)

For instance, if every element of $H$ has order $1, 2, 4, 8, 16,\ldots$ (any power of 2), then $H$ is a 2-subgroup. If every element of $H$ has order $1, 5, 25, 125,\ldots$ (any power of 5), then $H$ is a 5-subgroup. One does not say “4-subgroup” or “20-subgroup” because those numbers are not prime. Also, one can see that if $H$ is a $p$-subgroup for some prime $p$, then it is not a $q$-subgroup for any other prime $q$ unless $H= \{ e \}$.

If you click “tell me more” next to Subgroups in the Computations section of any group info page, you will see that the descriptions of the subgroups tell you which ones are $p$-subgroups.

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Preimage (of a subset under a morphism)

If a homomorphism $f$ maps a group $G$ to a group $G'$ and there is a subset $S$ of $G'$, we can find out what its “pre-image” is under the homomorphism $f$ by finding all the elements which $f$ maps into $S$. That is, the preimage of $S$ under $f$, sometimes written $f^{-1}[S]$, is the set $\left\{ a : f(a)\text{ is in }S \right\}$. Obviously this will be a subset of $G$. See also image.

Proper subgroup

A subgroup is proper if it is not the whole group.

Technically, by the definition of subgroup, every group is a subgroup of itself. But when we say “a proper subgroup” we mean subgroups that are actually smaller than the group we’re looking inside.

Quotient group

If $H$ is a subgroup of $G$, then we can sometimes make a group out of the cosets of $H$ as follows. The cosets of $H$ (let’s use the left cosets) are the sets $aH$ for various values of $a$. To define an operation on this collection, we let $aH$ times $bH$ equal $(ab)H$. One can prove that this operation is a valid group operation (as per the definition of a group) if and only if $H$ is a normal subgroup of $G$. In that case, the new group we just formed is called the quotient of $G$ by $H$, written $\frac{G}{H}$.

Multiplication tables and Cayley diagrams can both organize themselves by the cosets of a subgroup and then separate those cosets (or “chunk” them) to help you visualize the quotient operation. Refer to the documentation on the multiplication table interface or the Cayley diagram interface for more information on these features.

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Right cosets

See cosets.

Short exact sequence

An exact sequence is a chain of groups connected by homomorphisms such that the image of any one homomorphism in the chain is the kernel of the next homomorphism. A short exact sequence is one with only five groups in it, the first and last of which are both the trivial group. An example is shown below.

A short exact sequence is related to the quotient operation on groups. Let us call the four morphisms in a short exact sequence $id$, $e$, $q$, and $z$, from left to right, and the three middle groups in the sequence $A$, $B$, and $C$, also from left to right.

I call the first homomorphism $id$ (for “identity”) because it simply maps the one element in its domain (the identity element) to the identity element in $A$.

As per the definition above, the kernel of $e$ must be the image of $id$, which is simply the identity element. A nice theorem of group theory tells us that when a morphisms’ kernel is the identity, the whole morphism is injective; thus a copy of $A$ appears in $B$, that copy being the image of $e$ (which stands for “embedding,” a synonym for injection).

In turn, the kernel of $q$ must be the image of $e$, which is an isomorphic copy of $A$. This means that the map $q$ effectively zeroes out or removes $A$ from $B$. The First Isomorphism Theorem then implies that $q$ is a quotient map (hence the name) and that $\frac{B}{A} = \text{Im}(q)$.

But there’s more! Because the image of $z$ is only the identity element, its kernel is all of $C$. Therefore because this is an exact sequence, the image of $q$ must also be $C$. This means that the short exact sequence $A\to B\to C$ illustrates the fact that $\frac{B}{A}=C$.

You can see a short exact sequence illustrated for any normal subgroup of any group. Go to a group’s info page, to the Subgroups section under Computations, and click “tell me more.” Any normal subgroup will provide a link to a sheet illustrating the quotient via a short exact sequence.

Simple group

A simple group is a non-trivial group with no non-trivial, proper, normal subgroups. Among abelian groups, the only simple groups are the cyclic groups of prime order. The smallest non-abelian simple group is $A_5$.

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Solvable group, solvable decomposition

Solvable groups are important in Galois theory, which is too large a topic to embark on here. Briefly, Galois theory was invented to study which polynomials are solvable using ordinary arithmetic plus radicals. If you are interested in Galois theory, refer to an abstract algebra textbook or an online resource.

The roots of polynomials have symmetry that can be described by groups, called the Galois group of the polynomials. Evariste Galois (a 19th century mathematician) proved that you could tell by looking at these groups whether the polynomial was solvable. Groups that corresponded to solvable polynomials got the name “solvable groups.”

A group $G$ is solvable if there is a chain of groups $H_1, H_2, H_3, \ldots, H_n$ such that each group is a normal subgroup of the next one in the chain, the resulting quotient groups are all abelian, and the chain begins with the trivial group and ends with $G$. You can see a diagram illustrating this for any solvable group by looking under the Computations section of the group’s info page.

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Subgroup

If $S$ is a subset of the group $G$ (i.e. a subset of the set of elements of $G$) then we say $S$ is a subgroup if it is also a group under the operation of $G$.

A subset $S$ of a group $G$ may fail to be a subgroup in a few different ways; here are examples.

• If $S$ does not contain the identity element, it violates one of the criteria in the definition of a group.
• If $S$ contains an element but not that element’s inverse, it would violates another of those criteria.
• If $S$ contains two elements but not their product, then the binary operation of $G$ cannot be said to be a binary operation on $S$, because it maps some pairs from $S$ outside of $S$.

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Surjective, surjection

A function is surjective (or onto, or a surjection) from a set $A$ to a set $B$ if every element of $B$ is mapped to by some element of $A$. That is, if $b$ is in $B$, then there must be some $a$ in $A$ such that $f(a) = b$. A surjective homomorphism is sometimes called an epimorphism.

Sylow p-subgroup

A Sylow $p$-subgroup is a maximal $p$-subgroup; that is, no subgroup properly containing this one is still a $p$-subgroup.

If you click “tell me more” next to Subgroups in the Computations section of any group info page, you will see that the descriptions of the subgroups tell you which ones are Sylow $p$-subgroups.

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Symmetry objects

See objects of symmetry.

Trivial group/subgroup

The trivial group is the group with only one element. You can see its information here.

In every group, the set containing only the identity element is a subgroup and is called the trivial subgroup.