How many groups are there of order 7?

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Homework Help Overview

The discussion revolves around the number of groups of order 7, which is a topic in group theory, a branch of abstract algebra. The original poster seeks clarification to proceed with an assignment.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the order of the group being a prime number and the characteristics of group elements. There is a focus on the number of subgroups that can exist within a group of this order.

Discussion Status

Some participants have offered insights regarding the properties of groups of prime order, and there is an acknowledgment of the simplicity of the answer. However, the discussion does not reach a definitive consensus on the number of groups.

Contextual Notes

There is an underlying assumption that the participants are familiar with group theory concepts, such as group order and subgroups, which may not be explicitly stated in the thread.

Centurion1
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How many groups are there of order 7.

I just need to know this to continue with an assignment.

Thanks.
 
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Take a look at the group elements,

[tex]Z_{7} = \left\{ 1, a, a^{2},a^{3},a^{4},a^{5},a^{6}\right\}[/tex]
 
The crucial point is that 7 is a prime number. How many subgroups can a group of order 7 have? What does that tell you?
 
yes in hindsight this was such a duh answer.

it is one correct?
 
it is one correct?

Correct.
 

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