What is the difference between a Subgroup and a subset?

Solution 1:

Consider the group $\Bbb Z$ of integers under addition. The subset $S=\{1,2\}$ is not a subgroup. It isn’t a group at all: it isn’t closed under the operation (addition), since $2+2$ is not in $S$, and it doesn’t have an additive identity.

The set $E$ of all even integers, on the other hand, is a subset of $\Bbb Z$ that is a subgroup: it’s a group in its own right using the same operation, addition, as $\Bbb Z$.

If $G$ is any group, every subgroup of $G$ is by definition a subset of $G$, but as the example above shows, not every subset of $G$ need be a subgroup.

Solution 2:

Groups are sets with additional structure: a binary operation which has certain properties.

A subgroup is a subset which is also a group of its own, in a way compatible with the original group structure. But not every subset is a subgroup. To be a subgroup you need to contain the neutral element, and be closed under the binary operation, and the existence of an inverse.

For example, groups are never empty (they have a neutral element), so the empty set is always a subset but never a subgroup. The rational numbers are a subgroup of the real numbers, and a subset of the real numbers, whereas $\{0,1\}$ is a subset but not a subgroup, $1+1\neq 0$. Similarly, $\mathbb N$ is a subset of $\mathbb R$ which is not a group, despite being closed under addition, $1$ does not have an additive inverse (which is $-1$) in $\mathbb N$.

Solution 3:

You have laws that make a set $S$ become a group. You must have an operation $\star$ associated with this set which is closed under this operation (if $x,y\in S$ then $x \star y \in S$) and the operation must have these three properties :

1. If $a,b,c\in S$ then $a\star(b\star c)=(a\star b)\star c$
2. There exists a (unique) identity $e_S$ such that $a\star e_S=e_S\star a=a$ for every $a\in S$
3. For every $a\in S$, there exists a (unique) inverse $b$ such that $a\star b=b\star a=e_S$

If all these conditions are filled, then $S$ is called a group.

A subset of $S$ is just a subset of $S$ as a set.

But a subgroup of $S$ must have the three properties above.

Solution 4:

A subgroup $H$ of $G$ it's only a non-empty subset of $G$ such that $x - y \in H$ if $x,y \in H$, that's to say, a subset of $G$ with additional properties. For example, by definition the empty set is not a subgroup.