1 / 6

Group Theory Subgroups Analysis and Proof Steps

Explore subgroups of Klein-4 group and Z4 group. Follow a step-by-step proof for recognizing group characteristics.

wunzueta
Download Presentation

Group Theory Subgroups Analysis and Proof Steps

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Find all subgroups of the Klein 4-Group. How many are there? 1 2 3 4 5 6 7 8 9 10

  2. Find all subgroups of Z4 . How many are there? 1 2 3 4 5 6 7 8 9 10

  3. What is the first line in this proof? • Assume G is an abelian group. • Assume G is a cyclic group. • Assume a * b = b * a.

  4. What is the next line in this proof? • Then G is a subgroup of H. • Then G contains inverses. • Let a, b be any two elements in G. • Let H be any subgroup in G.

  5. What is the last line in this proof? • Thus G is abelian. • Thus H contains inverses. • Therefore H is cyclic. • Then G has primary order.

  6. What is the second to last line in this proof? • Then G is cyclic. • Then G has finite order. • Then H = <?> for some ? in G. • Then H has finite order.

More Related