Tour:Subgroup

This article adapts material from the main article: Subgroup

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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PREREQUISITES: Definition of group. Return to group if you do not remember this.
WHAT YOU NEED TO DO:

• Understand thoroughly, the definition in terms of closure of binary operation and the universal algebraic definition of subgroup
• Read, and make sense, of the other definitions
• Make sense of the examples given and use these to get some intuition about what being a subgroup means
  

WHAT YOU DO NOT NEED TO DO: Prove that all the definitions are equivalent. This will be covered in part two of the tour.

IF YOU ARE FOLLOWING ANOTHER PRIMARY TEXT: Compare the definition of subgroup given here, with the definition in your primary text

Definition

Definition in terms of closure under binary operation

This definition of subgroup corresponds to the textbook definition of group.

Let $G$ be a group. A subset $H$ of $G$ is termed a subgroup if the following two conditions hold:

• Whenever $a,b$ belong to $H$, the product $ab$ belongs to $H$.
• With this induced multiplication, $H$ becomes a group in its own right (i.e., it has an identity element, and every element has a two-sided inverse). Note that associativity in $H$ follows automatically from associativity in $G$.

The universal algebraic definition

This definition of subgroup corresponds to the universal algebraic definition of group.

Let $G$ be a group. A subset $H$ of $G$ is termed a subgroup if all the three conditions below are satisfied:

• Whenever $a,b$ belong to $H$, so does $ab$ (here $ab$ denotes the product of the two elements)
• $e$ belongs to $H$ (where $e$ denotes the identity element)
• Whenever $a$ belongs to $H$, so does $a^{-1}$ (the multiplicative inverse of $a$)

Definition via the subgroup condition

The equivalence of this definition with the earlier one is often called the subgroup condition. For full proof, refer: Sufficiency of subgroup condition

It has two forms (left and right):

• A subset of a group is termed a subgroup if it is nonempty and is closed under the left quotient of elements. In other words, a subset $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $a^{-1}b\in H$ whenever $a,b\in H$
• A subset of a group is termed a subgroup if it is nonempty and is closed under the right quotient of elements. In other words, a subset $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $ab^{-1}\in H$ whenever $a,b\in H$

Notation

If $H$ is a subgroup of $G$, we typically write $H\leq G$ or $G\geq H$. Some people also write $H\subseteq G$, but the latter notation is typically used for arbitrary subsets that need not be subgroups.

If $H$ is not equal to the whole of $G$, we say that $H$ is a proper subgroup of $G$, and this is sometimes denoted by $H<G$ or $G>H$.

Examples

Examples in abelian groups

If we consider the abelian group $(\mathbb {R} ,+)$ (reals under addition) then the group of integers $(\mathbb {Z} ,+)$ is a subgroup of this group. Similarly, the group of rational numbers ($\mathbb {Q} ,+)$) is an example of a subgroup of the group of reals.

On the other hand, the set of positive integers is not a subgroup of the group of integers, although it is closed under the group operation. This is because the additive inverse (or negative) of a positive integer isn't a positive integer.

PONDER (WILL BE EXPLORED LATER IN THE TOUR):

• Over why the different definitions of subgroup are equivalent.
• Over what parts of the definition of group are needed to prove the equivalence of definitions

WHAT'S MORE: Some more definitions, examples, and general information about subgroups. Some of it may use terminology that you haven't encounutered so far; ignore those parts.

Definition in terms of injective homomorphisms

A subgroup of a group can also be defined as another abstract group along with an injective homomorphism (or embedding) from that abstract group to the given group. Here, the other abstract group can be naturally identified via its image under the homomorphism, which is the subgroup in a more literal sense.

Often, when we want to emphasize the subgroup not just as an abstract group but in its role as a subgroup, we use the term embedding and think of it as an injective homomorphism.

Equivalence of definitions

Further information: Equivalence of definitions of subgroup, Sufficiency of subgroup criterion

The equivalence of the two definitions (the definition in terms of closure under binary operation and the universal algebraic definition) relies on the following two facts:

1. If a subset of a group is closed under multiplication and has an identity element, then that identity element must equal the identity element of the group (this relies on the cancellation property in groups).
2. If a subset of a group allows for multiplicative inverses, then the multiplicative inverses inside the subset are the same as the multiplicative inverses inside the whole group

The equivalence with the definition arising from the subgroup criterion is based on a short and elegant argument, refer sufficiency of subgroup criterion.

The equivalence with the definition in terms of injective homomorphism relies on viewing the subgroup as a group in its own right and its inclusion in the whole group as an injective homomorphism.

Equivalence of subgroups

Given a subgroup $H_{1}\leq G_{1}$ and a subgroup $H_{2}\leq G_{2}$, we say that these two subgroups are equivalent if there is an isomorphism $\sigma$ from $G_{1}$ to $G_{2}$ such that $H_{1}$ maps to $H_{2}$ under that isomorphism.

In particular, if $G_{1}=G_{2}=G$, then $H_{1}$ and $H_{2}$ are equivalent as subgroups if there is an automorphism of $G$ under which $H_{1}$ maps to $H_{2}$ (subgroups equivalent in this sense are termed automorphic subgroups or automorphs -- sometimes, stronger notions of equivalence, such as being conjugate subgroups, are also useful).

This notion of equivalence of subgroups is important when dealing with and defining the notion of subgroup property.

Examples in non-abelian groups

Consider the group of all permutations of the set of elements $\{1,2,3,\ldots ,n\}$. This is termed the symmetric group on $n$ elements. The group of permutations that fix $n$ is a subgroup of this group.

Subgroups usually arise as elements of the group satisfying some additional conditions, where that condition is preserved under taking inverses, preserved under multiplication, and satisfied by the identity element.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
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