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Finite Abelian Groups
In our investigation of cyclic groups we found that every group of prime order was isomorphic to {\mathbb Z}_p, where p was a prime number.
Finite Abelian Groups
In our investigation of cyclic groups we found that every group of prime order was isomorphic to \({\mathbb Z}_p\), where \(p\) was a prime number. We also determined that \({\mathbb Z}_{mn} \cong {\mathbb Z}_m \times {\mathbb Z}_n\) when \(\gcd(m, n) =1\). In fact, much more is true. Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order; that is, every finite abelian group is isomorphic to a group of the type \[\begin{aligned}\end{aligned}\], where each \(p_k\) is prime (not necessarily distinct).
First, let us examine a slight generalization of finite abelian groups. Suppose that \(G\) is a group and let \(\{ g_i\}\) be a set of elements in \(G\), where \(i\) is in some index set \(I\) (not necessarily finite). The smallest subgroup of \(G\) containing all of the \(g_i\)'s is the subgroup of \(G\) generated by the \(g_i\)'s. If this subgroup of \(G\) is in fact all of \(G\), then \(G\) is generated by the set \(\{g_i : i \in I \}\). In this case the \(g_i\)'s are said to be the generators of \(G\). If there is a finite set \(\{ g_i : i \in I \}\) that generates \(G\), then \(G\) is finitely generated.
Example
Obviously, all finite groups are finitely generated. For example, the group \(S_3\) is generated by the permutations \((1 \, 2)\) and \((1 \, 2 \,3)\). The group \({\mathbb Z} \times {\mathbb Z}_n\) is an infinite group but is finitely generated by \(\{ (1,0), (0,1) \}\).
Example
Not all groups are finitely generated. Consider the rational numbers \({\mathbb Q}\) under the operation of addition. Suppose that \({\mathbb Q}\) is finitely generated with generators \(p_1/q_1, \ldots, p_n/q_n\), where each \(p_i/q_i\) is a fraction expressed in its lowest terms. Let \(p\) be some prime that does not divide any of the denominators \(q_1, \ldots, q_n\). We claim that \(1/p\) cannot be in the subgroup of \({\mathbb Q}\) that is generated by \(p_1/q_1, \ldots, p_n/q_n\), since \(p\) does not divide the denominator of any element in this subgroup. This fact is easy to see since the sum of any two generators is \[\begin{aligned}\end{aligned}\].
The reason that powers of a fixed \(g_i\) may occur several times in the product is that we may have a nonabelian group. However, if the group is abelian, then the \(g_i\)s need occur only once. For example, a product such as \(a^{-3} b^5 a^7\) in an abelian group could always be simplified (in this case, to \(a^4 b^5\)).
Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.
Manje Akukho kalkuli yokuxazulula le nto, kodwa amaphuzu ayo anga kalkuli. Zama enye ngezansi, noma ubhale yakho.
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
Imibuzo abantu bebuza
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
Amangxenye wale khasi aguqulwe kusuka Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). I-condensed futhi iphinde icaciswe lapha; amaphutha awusizo.
Okuningi Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula