آموزش ریاضی توسط استاد صدیقه حافظی
گروه طراحی قالب من گروه طراحی قالب من گروه طراحی قالب من گروه طراحی قالب من گروه طراحی قالب من
درباره وبلاگ



مدیر وبلاگ : صدیقه حافظی
نویسندگان

Integers, rationals and reals

  • The integers under addition form a group. The identity element of the group is 0, and the additive inverse is just the usual negative. In fact, the group of integers is an Abelian group: addition is commutative for integers.
  • The rational numbers under addition form a group. The identity element of the group is 0, and the additive inverse is just the usual negative. This group is Abelian, and the integers form a subgroup.
  • The real numbers also form a group under addition. The rational numbers form a subgroup of the group of real numbers, and the integers form a smaller subgroup.
  • The nonzero rational numbers under multiplication form a group. The identity element for this group is 1. This group is also Abelian.

More generally, given any field, the field is a group under addition, and the nonzero elements of the field form a group under multiplication.

Some non-examples of groups are:

  • The natural numbers under addition: There is no additive identity and there are no additive inverses.
  • The nonzero integers under multiplication: The nonzero integers under multiplication have a multiplicative identity (namely 1). Hence, they form a monoid. But not every nonzero integer has an integer as its multiplicative inverse. In fact, the only invertible elements are \pm 1.

برای ادامه مثال ها روی ادامه مطلب کلیک کنید...

Modular arithmetic: groups from number theory

Further information: cyclic group,cyclic implies AbelianOne of the ways of constructing finite groups is to look at integers modulo a given nonzero integer n. By integers modulo n, we mean that we are looking at the group of integers, modulo the equivalence relation of differing by a multiple of n. For instance, modulo 2, there are exactly two equivalence classes of numbers: the even numbers and odd numbers. Thus, the group of integers modulo 2, termed the cyclic group of order two, has exactly two elements, one corresponding to the collection of even numbers and one corresponding to the collection of odd numbers. These are typically represented as 0 and 1. The group operation is then given by:

0 + 0 = 0, \qquad 0 + 1 = 1, \qquad 1 + 0 = 1,\qquad 1 + 1 = 0

Similarly, modulo 4, there are four equivalence classes of numbers: the multiples of 4, the numbers that leave a remainder of 1 modulo 4, the numbers that leave a remainder of 2 modulo 4, and the numbers that leave a remainder of 3 modulo 4.

The equivalence classes of numbers modulo n form a group under addition. For instance, whenever we add a number that is 1 mod 4 and a number that is 2 mod 4, we get a number that is 3 mod 4.

For convenience, we represent an equivalence class modulo n by the smallest nonnegative integer representative. So the four equivalence classes modulo 4 are represented by the elements 0,1,2,3 respectively, and while adding, we reduce the sum modulo 4 (so 2 + 3 = 1).

Groups that are obtained in this way are termed cyclic groups.

Another way of viewing cyclic groups is as quotients of the group of integers by a normal subgroup.

Permutations: groups from functions

Further information: symmetric group

A permutation of a set S is a bijective map from S to itself. The symmetric group on a set is the set of all permutations on it, where:

  • The product of two permutations is composition. If f and g are permutations, their product is the map x \mapsto f(g(x))
  • The identity map is the identity element
  • The inverse of a permutation is its inverse as a function

The symmetric groups are important examples of non-Abelian groups: in fact the symmetric group on a set of size at least three, is always non-Abelian. Moreover, it is surprisingly true that every finite group occurs as a subgroup of the symmetric group, so symmetric groups are subgroup-rich. Further information: Cayley's theorem

Transformations: groups from geometry

Further information: Groups as symmetry

Yet another example of groups comes from geometry. In the geometric context, groups occur as symmetries of structure that preserve certain geometric properties. As in the case of the groups of permutations, the multiplication is by composition, the identity element is the identity map, and the inverse map sends a transformation to its inverse function.

For instance, the rotations of the plane about a point form a group (in fact, an Abelian group). Here, each rotation is described by a (signed) angle of rotation, and to add two rotations , we add their angles. However, the angles are viewed modulo 2 \pi, so the group of rotations is the group of real numbers quotiented by the equivalence relation of differing by 2\pi.

Non-Abelian groups also arise in geometry. In fact, rotations about a fixed point in three-dimensional space form a non-Abelian group. Rotations around different axes do not, in general, commute.

Constructing groups with certain subgroup structures

One question of interest is: can we find groups with certain structural attributes? Questions about existence of groups are usually very complicated, and we consider here a very simple version of these questions.

Every nontrivial group has at least two subgroups: the trivial subgroup and the group itself. What are the groups for which there are no other subgroups?

To solve this, we use the following ideas:

  • Given any nontrivial element of the group, we can consider the cyclic subgroup generated by that. This is the smallest subgroup containing that element.
  • If the group has no nontrivial proper subgroup, then the cyclic subgroup generated by that element must be the whole group. In particular, the group must be cyclic.
  • So the group must either be the group of integers, or it must be the group of integers modulo n. But the group of integers has lots of subgroups (for instance, multiples of n). The group of integers modulo n has proper nontrivial subgroups if n is not prime.
  • So the only possibility for a group with no proper nontrivial subgroup is a cyclic group of prime order
  • Conversely, any cyclic group of prime order has no proper nontrivial subgroup




نوع مطلب : مقاله، 
برچسب ها :
لینک های مرتبط :
صدیقه حافظی
دوشنبه 18 دی 1391
شنبه 28 اردیبهشت 1398 11:25 ق.ظ
best buy deals canada http://viagralim.us best buy deals canada !
I absolutely love your blog and find almost all of your post's to be just what I'm looking for. Do you offer guest writers to write content available for you? I wouldn't mind writing a post or elaborating on some of the subjects you write concerning here. Again, awesome blog!
شنبه 22 دی 1397 03:13 ب.ظ
кредит на карту онлайн
кредит онлайн круглосуточно
кредит через интернет по паспорту
взять кредит через интернет на карту
получить кредит онлайн на карту
شنبه 22 دی 1397 02:56 ب.ظ
онлайн кредит
оформление кредита онлайн
кредиты онлайн на карту
получить кредит на карту
кредит на карту срочно
شنبه 22 اردیبهشت 1397 02:23 ق.ظ
кредит без фото
кредит срочно без отказов онлайн
кредит онлайн 24 7
круглосуточные займы онлайн на
карту
онлайн кредит без отказов
займ онлайн украина на карту
быстрый займ на карту онлайн круглосуточно
سه شنبه 11 اردیبهشت 1397 05:55 ب.ظ
купить резинки для фитнеса в интернет магазинах спорт
розетка фитнес резинки
фитнес резинки купить днепр
фитнес резинка размеры
резинки для фитнеса киев купить
سه شنبه 21 فروردین 1397 03:05 ب.ظ
Lengthy necklaces, short necklaces, or necklaces which keep very near your neck, better know as chokers, are
the completely different styles of necklaces.
یکشنبه 19 فروردین 1397 02:36 ق.ظ
Have actually been taking little over a month.
جمعه 17 فروردین 1397 04:24 ب.ظ
Have been taking little over a month.
جمعه 17 فروردین 1397 10:56 ق.ظ
Have actually been taking little over a month.
جمعه 17 فروردین 1397 07:57 ق.ظ
The models are fine.
جمعه 17 فروردین 1397 04:20 ق.ظ
Make your traffic creating investment worthwhile's weight in silver.
جمعه 17 فروردین 1397 03:51 ق.ظ
It is a good example of when a simple moissanite ring would have sufficed, to not point out saved
plenty of nice individuals complications (and severed limbs).
شنبه 5 اسفند 1396 07:52 ق.ظ
من این وب سایت را از طریق پسر عمویم پیشنهاد دادم. من مثبت نیستم که آیا این پست از طریق او به عنوان هیچ کس دیگری نوشته شده است
چنین تقریبی را به من نزدیک می کند. شما هستید
حیرت آور! متشکرم!
دوشنبه 30 بهمن 1396 07:10 ق.ظ
هی، وبلاگ بسیار خوب!
جمعه 17 آذر 1396 04:21 ق.ظ
This post will assist the internet visitors for building
up new website or even a blog from start to end.
شنبه 1 مهر 1396 08:32 ب.ظ
Hi to all, it's truly a nice for me to pay a visit this website, it consists of important Information.
دوشنبه 13 شهریور 1396 09:24 ب.ظ
Do you have a spam issue on this blog; I also am a blogger, and I was wondering
your situation; we have created some nice practices and we are looking to swap solutions with other
folks, why not shoot me an e-mail if interested.
دوشنبه 13 شهریور 1396 09:17 ب.ظ
I have to thank you for the efforts you have put in writing this website.
I really hope to see the same high-grade
content by you later on as well. In fact, your creative
writing abilities has inspired me to get my very own site now
;)
دوشنبه 13 شهریور 1396 08:46 ب.ظ
Thanks designed for sharing such a good opinion, piece of
writing is good, thats why i have read it fully
دوشنبه 13 شهریور 1396 08:02 ب.ظ
Incredible quest there. What happened after? Take care!
دوشنبه 16 مرداد 1396 04:30 ق.ظ
I just couldn't depart your website before suggesting that I extremely loved the
standard information a person supply in your guests? Is gonna be again continuously in order to inspect new posts
شنبه 7 مرداد 1396 12:41 ق.ظ
It's remarkable to go to see this web site and reading the views of all
mates concerning this article, while I am also eager of getting experience.
شنبه 7 مرداد 1396 12:10 ق.ظ
Hurrah! After all I got a weblog from where
I know how to really get valuable data regarding my study and knowledge.
جمعه 6 مرداد 1396 11:04 ب.ظ
Hey I know this is off topic but I was wondering if you knew of any
widgets I could add to my blog that automatically tweet my newest twitter updates.
I've been looking for a plug-in like this for quite some time and was hoping maybe
you would have some experience with something like this.
Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.
جمعه 6 مرداد 1396 10:11 ب.ظ
Because the admin of this web page is working, no question very rapidly it will be famous,
due to its quality contents.
جمعه 6 مرداد 1396 09:53 ب.ظ
Generally I do not read article on blogs, but I wish to say
that this write-up very compelled me to try and do so! Your writing style has been surprised me.
Thank you, quite nice post.
جمعه 6 مرداد 1396 08:38 ب.ظ
It's going to be ending of mine day, except before end I am reading this wonderful
post to increase my knowledge.
یکشنبه 25 تیر 1396 09:33 ب.ظ
Hi there to every body, it's my first pay a visit of this website; this weblog consists of awesome and genuinely excellent data designed for visitors.
چهارشنبه 7 تیر 1396 09:02 ب.ظ
Keep this going please, great job!
شنبه 3 تیر 1396 08:12 ب.ظ
بسیار قلب از خود نوشتن در حالی که
صدایی دلنشین در آیا نه حل و فصل بسیار خوب با
من پس از برخی از زمان. جایی در سراسر
جملات شما موفق به من مؤمن اما تنها برای کوتاه در حالی
که. من با این حال مشکل خود را با جهش در
منطق و یک ممکن است را خوب به
پر همه کسانی معافیت. اگر شما در واقع که می توانید انجام من خواهد مطمئنا تا پایان مجذوب.
 
لبخندناراحتچشمک
نیشخندبغلسوال
قلبخجالتزبان
ماچتعجبعصبانی
عینکشیطانگریه
خندهقهقههخداحافظ
سبزقهرهورا
دستگلتفکر


نمایش نظرات 1 تا 30


آمار وبلاگ
  • کل بازدید :
  • بازدید امروز :
  • بازدید دیروز :
  • بازدید این ماه :
  • بازدید ماه قبل :
  • تعداد نویسندگان :
  • تعداد کل پست ها :
  • آخرین بازدید :
  • آخرین بروز رسانی :
امکانات جانبی
ساخت وبلاگ در میهن بلاگ

شبکه اجتماعی فارسی کلوب | اخبار کامپیوتر، فناوری اطلاعات و سلامتی مجله علم و فن | ساخت وبلاگ صوتی صدالاگ | سوال و جواب و پاسخ | رسانه فروردین، تبلیغات اینترنتی، رپرتاژ، بنر، سئو