-412.7.3 decide if cosets of H are the same

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In summary, the cosets of H that are the same are those that have the same shift (i.e. 11+H and 17+H are the same, as are 5+H and -1+H, etc).
  • #1
karush
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Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$.
$\textit{ Use }$
$$aH=bH \textit{ or }aH\cap bH=\oslash$$
$\textit{then..}$
$$aH = bH \textit{ iff } (b-a) \textit{ is in } \textit{H}$$
to decide whether or not the following cosets of H are the same.
$\textsf{a. 11 + H and 17 + H}$
$\textsf{b. -1 + H and 5 + H}$
$\textsf{c. 7 + H and 23 + H}$ok not sure what the official method of this would be but for a. 11+6=17 so the coset would would be just a shift over 6 places. the same shift seens to be true for b and c however the beginning numbers would be different

however the book says that c is no
 
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  • #2
karush said:
Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$.
$\textit{ Use }$
$$aH=bH \textit{ or }aH\cap bH=\oslash$$
$\textit{then..}$
$$aH = bH \textit{ iff } (b-a) \textit{ is in } \textit{H}$$
to decide whether or not the following cosets of H are the same.
$\textsf{a. 11 + H and 17 + H}$
$\textsf{b. -1 + H and 5 + H}$
$\textsf{c. 7 + H and 23 + H}$ok not sure what the official method of this would be but for a. 11+6=17 so the coset would would be just a shift over 6 places. the same shift seens to be true for b and c however the beginning numbers would be different

however the book says that c is no
$H = \{\ldots -9;-6;-3;\ 0;\ 3;\ 6;\ 9 \ldots\}$

For a.: $11+H = \{\ldots 2;\ 5;\ 8;\ 11;\ 14;\ 17;\ 20;\ \ldots\},\quad 17+H = \{\ldots 8;\ 11;\ 14;\ 17;\ 20;\ 23;\ 25 \ldots\}$
Those two sets are the same. The numbers in the list have just been shifted by two places to get from $11+H$ to $17+H$.

For c.: $7+H = \{\ldots -2;\ 1;\ 4;\ 7;\ 10;\ 13;\ 16;\ 19;\ 22;\ 25;\ 28 \ldots\},\quad 23+H = \{\ldots 14;\ 17;\ 20;\ 23;\ 26 \ldots\}$
Those two sets are not the same. The numbers in them do not coincide at all.

Can you see what condition the shifts must satisfy in order for the two cosets to be the same?
 
  • #3
Yes: your "method" works. 11+ H and 17+ H are the same because 17- 11= 6 is a multiple of 3; -1+ H and 5+ H are the same because 5- (-1)= 6 is a multiple of 3; and 7+ H and 23+ H are NOT the same because 23- 7= 16 is NOT a multiple of 3.
 
  • #4

FAQ: -412.7.3 decide if cosets of H are the same

1. What is a coset?

A coset is a subset of a group that is formed by multiplying an element of the group by a specific element.

2. How do you determine if two cosets of H are the same?

To determine if two cosets of H are the same, you can multiply both cosets by an element of H and see if the resulting sets are equal. If they are, then the two cosets are the same.

3. What is the purpose of determining if cosets of H are the same?

Determining if cosets of H are the same helps to understand the structure and properties of the group. It also allows for simplification and easier computation in certain group operations.

4. Is it possible for cosets of H to be the same in one group and different in another?

Yes, it is possible for cosets of H to be the same in one group and different in another. The properties and structure of the group can affect the cosets and their equality.

5. Can cosets of H be the same if H is not a subgroup?

No, cosets of H can only be the same if H is a subgroup of the group. This is because the definition of a coset involves multiplying elements of the group by elements of H, which can only be done if H is a subgroup.

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