**Basic Properties of Subgroups**

- A subset
*H*of the group*G*is a subgroup of*G*if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever*a*and*b*are in*H*, then*ab*and*a*−1 are also in*H*. These two conditions can be combined into one equivalent condition: whenever*a*and*b*are in*H*, then*ab*−1 is also in*H*.) In the case that*H*is finite, then*H*is a subgroup if and only if*H*is closed under products. (In this case, every element*a*of*H*generates a finite cyclic subgroup of*H*, and the inverse of*a*is then*a*−1 =*a**n*− 1, where*n*is the order of*a*.) - The above condition can be stated in terms of a homomorphism; that is,
*H*is a subgroup of a group*G*if and only if*H*is a subset of*G*and there is an inclusion homomorphism (i.e., i(*a*) =*a*for every*a*) from*H*to*G*. - The identity of a subgroup is the identity of the group: if
*G*is a group with identity*e*_{G}, and*H*is a subgroup of*G*with identity*e*_{H}, then*e*_{H}=*e*_{G}. - The inverse of an element in a subgroup is the inverse of the element in the group: if
*H*is a subgroup of a group*G*, and*a*and*b*are elements of*H*such that*ab*=*ba*=*e*_{H}, then*ab*=*ba*=*e*_{G}. - The intersection of subgroups
*A*and*B*is again a subgroup. The union of subgroups*A*and*B*is a subgroup if and only if either*A*or*B*contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity. - If
*S*is a subset of*G*, then there exists a minimum subgroup containing*S*, which can be found by taking the intersection of all of subgroups containing*S*; it is denoted by <*S*> and is said to be the subgroup generated by*S*. An element of*G*is in <*S*> if and only if it is a finite product of elements of*S*and their inverses. - Every element
*a*of a group*G*generates the cyclic subgroup <*a*>. If <*a*> is isomorphic to**Z**/*n***Z**for some positive integer*n*, then*n*is the smallest positive integer for which*a**n*=*e*, and*n*is called the*order*of*a*. If <*a*> is isomorphic to**Z**, then*a*is said to have*infinite order*. - The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup
*generated by*the set-theoretic union of the subgroups, not the set-theoretic union itself.) If*e*is the identity of*G*, then the trivial group {*e*} is the minimum subgroup of*G*, while the maximum subgroup is the group*G*itself.

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### Famous quotes containing the words basic and/or properties:

“When you realize how hard it is to know the truth about yourself, you understand that even the most exhaustive and well-meaning autobiography, determined to tell the truth, represents, at best, a guess. There have been times in my life when I felt incredibly happy. Life was full. I seemed productive. Then I thought,”Am I really happy or am I merely masking a deep depression with frantic activity?” If I don’t know such *basic* things about myself, who does?”

—Phyllis Rose (b. 1942)

“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)