Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.
Cyclic Subgroups
Often a subgroup will depend entirely on a single element of the group;
that is, knowing that particular element will allow us to compute any other element in the subgroup.
Example
Suppose that we consider \(3 \in {\mathbb Z}\) and look at all multiples
(both positive and negative)
of \(3\).
As a set, this is
\[\begin{aligned}\end{aligned}\].
It is easy to see that \(3 {\mathbb Z}\) is a subgroup of the integers.
This subgroup is completely determined by the element \(3\) since we can obtain all of the other elements of the group by taking multiples of \(3\).
Every element in the subgroup is
generated by \(3\).
Example
If \(H = \{ 2^n : n \in {\mathbb Z} \}\),
then \(H\) is a subgroup of the multiplicative group of nonzero rational numbers,
\({\mathbb Q}^*\).
If \(a = 2^m\) and \(b = 2^n\) are in \(H\),
then \(ab^{-1} = 2^m 2^{-n} = 2^{m-n}\) is also in \(H\).
By ,
\(H\) is a subgroup of \({\mathbb Q}^*\) determined by the element \(2\).
For \(a \in G\), we call \(\langle a \rangle\) the
cyclic subgroup
generated by \(a\).
If \(G\) contains some element \(a\) such that \(G = \langle a \rangle\),
then \(G\) is a cyclic group.
In this case \(a\) is a generator of \(G\).
If \(a\) is an element of a group \(G\),
we define the order
of \(a\) to be the smallest positive integer \(n\) such that \(a^n= e\),
and we write \(|a| = n\).
\(|a|\)
the order of an element \(a\)
If there is no such integer \(n\),
we say that the order of \(a\) is infinite and write
\(|a| = \infty\) to denote the order of \(a\).
Example
Notice that a cyclic group can have more than a single generator.
Both \(1\) and \(5\) generate \({\mathbb Z}_6\);
hence, \({\mathbb Z}_6\) is a cyclic group.
Not every element in a cyclic group is necessarily a generator of the group.
The order of \(2 \in {\mathbb Z}_6\) is \(3\).
The cyclic subgroup generated by \(2\) is \(\langle 2 \rangle = \{ 0, 2, 4 \}\).
The groups \({\mathbb Z}\) and \({\mathbb Z}_n\) are cyclic groups.
The elements \(1\) and \(-1\) are generators for \({\mathbb Z}\).
We can certainly generate \({\mathbb Z}_n\) with \(1\) although there may be other generators of \({\mathbb Z}_n\),
as in the case of \({\mathbb Z}_6\).
Example
The group of units, \(U(9)\),
in \({\mathbb Z}_9\) is a cyclic group.
As a set, \(U(9)\) is \(\{ 1, 2, 4, 5, 7, 8 \}\).
The element \(2\) is a generator for \(U(9)\) since
\[\begin{aligned}2^1 & = 2 \qquad 2^2 = 4 \\ 2^3 & = 8 \qquad 2^4 = 7 \\ 2^5 & = 5 \qquad 2^6 = 1\end{aligned}\].
Example
Not every group is a cyclic group.
Consider the symmetry group of an equilateral triangle \(S_3\).
The multiplication table for this group is .
The subgroups of \(S_3\) are
Notice that every subgroup is cyclic;
however, no single element generates the entire group.
Subgroups of Cyclic Groups
We can ask some interesting questions about cyclic subgroups of a group and subgroups of a cyclic group.
If \(G\) is a group, which subgroups of \(G\) are cyclic?
If \(G\) is a cyclic group,
what type of subgroups does \(G\) possess?
اب آپ
هيک کليچر اس کو حل نہیں کر سکتا ، ليکن اس کے ٹکڑے حسابي هے. نيچے سے کو ئي کوشش کريں يا خود کو ٹائپ کريں
آپ کے اپنے کام رکھیں
مفت اکاؤنٹ تمام درسوں پر نوٹ شامل کر تا هے ، آپ نے جو ختم کر يا هے اس کا ريکورڈ ، آپ کے حل کيے گئے مسئلے ایک جگہ پر اور آپ اس صفحے کے بارے ميں سوال کر سکتے هيں تو تيرچر ريتھ ميں خود تمام لوگوں کے ليے کھلا هے ، خواہ آپ داخل هے يا نہيں
تمام تعريف کے ليے ، تصوير کے ليے اور اس ميں ہر حروف کا مطلب کے ليے کو ئي علامت کو ٹک کريں
ہر حصے کا مطلب کیا ہے
اس کو تصور کریں
جاننے کے قابل
مثالیں
اس وضاحت میں بھی استعمال کیا جاتا ہے
سوالات لوگ پوچھتے ہیں
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.