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Subgroups in .NET Writer QR Code 2d barcode in .NET Subgroups




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8.2 Subgroups use .net framework qr barcode creation topaint qr code with .net Viual Cshap Subgroups are subsets of gro QR Code for .NET ups which are groups "in their own right". A subset H of a group G is said to be a subgroup if, with the same operation as that used in G, it is a group.

That is, if H contains the identity element e E G, if H contains inverses of all elements in it, and if H contains products of any two elements in it, then H is a subgroup. (The associativity of the operation is assured since the operation was assumed associative for G itself to be a group.) .

Another paraphrase: if e E H, and if for all h E H the inverse h -1 is also in H, and if for all hit h2 E H the product hlh2 is again in H, then H is a subgroup ofG. Another cute paraphrase is: if e E H, and if for all hi, h2 E H the product hlh21 is again in H, then H is a subgroup of G. (If we take hI = e, then the latter condition assures the existence of inverses! And so on.

) In any case, one usually says that H is closed under inverses and closed under the group operation. (These two conditions are independent of each other.) For example, the collection of all even integers is a subgroup of the additive group of integers.

More generally, for fixed integer m, the collection H of all multiples of m is a subgroup of the additive group of integers. To check this: first,. 8 . Groups the identity 0 is a multiple of m, so 0 E H. And for any two integers x, y divisible by m, write x = ma and y = mb for some integers a, b. Then using the "cute" paraphrase, we see that.

x-y=ma-mb=m(a-b) E H so H is closed under inverses and under the group operation. Thus, it is a subgroup of Z. 8.3 Lagrange"s Theorem The theorem of this section is the simplest example of the use of group theory as structured counting. Although the discussion of this section is completely abstract, it gives the easiest route to (the very tangible) Euler"s theorem proven as a corollary below. A finite group is simply a group which is also finite.

The order of a finite group is the number of elements in it. Sometimes the order of a group G is written as IGI. Throughout this section we will write the group operation simply as though it were ordinary multiplication.

. Theorem: (Lagrange) Let G be a finite group. Let H be a subgroup of G. Then the order of H dividep the o rder of G. For the proof we need some other ideas which themselves will be reused later. For subgroup H of a group G, and for 9 E G, the left coset of H by 9 or left translate of H by 9 is gH = {gh : h E H} The notation gH is simply shorthand for the right-hand side.

Likewise, the right coset of H by 9 or right translate of H by 9 is. H 9 = {hg : h E H}. Proof: First, we will prove that the collection of all left cosets of H is a partition of G, meaning that every element of G lies in some left coset of H, and if two left cosets xH and yH have non-empty intersection then actually xH = yH. (Note that this need not imply x = y.) Certainly x = x e E xH, so every element of G lies in a left coset of H.

Now suppose that xHnyH =I- > for x,y E G. Then for some hl,h2 E H we have xh 1 = yh 2. Multiply both sides of this equality on the right by h;l to obtain.

The right-hand side of this is (yh 2)h21 = y(h2h;1) (by ass ociativity) = y . e (by property of inverse) = y (by property of e). 8.3 Lagrange"s Theorem Let z .NET QR Code JIS X 0510 = hIh2I for brevity.

By associativity in G,.
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