Let h and k be subgroups of a group g, and let

Ответ:

slimt69561

08.01.2022

Yes, double cosets partition G.

Step-by-step explanation:

We are going to define a relation over the elements of G.

Let $x,y\in G$ . We say that $x\sim y$ if, and only if, $y\in HxK$ , or, equivalently, if y=hxk , for some $h\in H, k\in K$ .

This defines an equivalence relation over G, that is, this relation is reflexive, symmetric and transitive:

Reflexivity: ( $x\sim x$ for all $x\in G$ .) Note that we can write x=exe

, where

is the identity element, so $e\in H,K$ and then $x\in HxK$ . Therefore, $x\sim x$ .Symmetry: (If $x\sim y$ then $y\sim x$ .) If $x\sim y$ then y=hxk

for some $h\in H$ and $k\in K$ . Multiplying by the inverses of h and k we get that $x=h^{-1}yk^{-1}$ and is known that $h^{-1}\in H$ and $k^{-1}\in K$ . This means that $x\in HyK$ or, equivalently, $y\sim x$ .Transitivity: (If $x\sim y$ and $y\sim z$ , then $x\sim z$ .) If $x\sim y$ and $y\sim z$ , then there exists $h_1,h_2\in H$ and $k_1,k_2\in K$ such that y=h_1xk_1

and

. Then, $\\ z=h_2yk_2=h_2(h_1xk_1)k_2=(h_2h_1)x(k_1k_2)=h_3xk_3$ where $h_3=h_2h_1\in H$ and $k_3=k_1k_2\in K$ . Consequently, $z\sim x$ .

Now that we prove that the relation " $\sim$ " is an equivalence over G, we use the fact that the different equivalence classes partition G. Since the equivalence classes are defined by $[x]=\{y\in G\colon x\sim y\} =\{y\in G \colon y=hgk\ \text{for some } h\in H, k\in K \}=HxK$ , then we're done.

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Ответ:

ecob35

18.04.2023

Mathematics

This question is incomplete, here is the complete question:

Suppose in the next year, 2007, College D's expenses and enrollment remain about the same, but in addition to their current revenues, they receive an additional $50,000,000 grant. This would allow them to reduce average tuition by how much?

A) $1388.89

B) $3571.43

C) $5555.56

D) $9500.00

E) $25888.89

number of students = 36,000

A) $

1388.89

Step-by-step explanation:

the college received additional grant which is $50,000,000

and the number of students is 36,000,

and we also know that expenses and enrollment remained the same.

So if we have more money (grants) and nothing changed (expenses remain the same)

dividing the grant by the number of students will show just how much the average tuition fee would be reduced

therefore R = G/n

R = 50,000,000 / 36000

R = 1,388.888 ≈ $1388.89

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Let h and k be subgroups of a group g, and let g be an element of g. the set $\math hgk = \{x \in g \mid x = hgk \text {for some} h \in h, k \in k\}$ is called a double coset. do the double cosets partition g?

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Let h and k be subgroups of a group g, and let g be an element of g. the set is called a double coset. do the double cosets partition g?

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Let h and k be subgroups of a group g, and let g be an element of g. the set $\math hgk = \{x \in g \mid x = hgk \text {for some} h \in h, k \in k\}$ is called a double coset. do the double cosets partition g?