What Group Contains Elements a and b in Abstract Algebra?

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SUMMARY

This discussion focuses on identifying groups in abstract algebra that contain elements a and b with specific properties. The first question seeks groups where the orders of a and b are both 2, and the order of their product ab can be 3, 4, or 5. The second question involves determining the proper subgroup H of the integers Z under addition, given that H contains the elements 18, 30, and 40. The solution requires understanding group properties and subgroup generation.

PREREQUISITES
  • Understanding of group theory concepts, specifically group orders.
  • Knowledge of subgroup properties in the context of integers under addition.
  • Familiarity with the concept of proper subgroups.
  • Basic skills in abstract algebra notation and terminology.
NEXT STEPS
  • Research the properties of groups with elements of specific orders, particularly in finite groups.
  • Study the structure of subgroups in additive groups, focusing on integer subgroups.
  • Learn about the classification of groups based on element orders and their products.
  • Explore examples of groups that satisfy specific conditions, such as the symmetric group S3.
USEFUL FOR

Students and educators in abstract algebra, mathematicians exploring group theory, and anyone seeking to deepen their understanding of group structures and subgroup identification.

ShengyaoLiang
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1)
find a group that contains elements a and b such that ︱a︱=︱b︱= 2 and

a) ︱ab︱ = 3 b) ︱ab︱=4 c) ︱ab︱=5

2)
suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40.
determine H?


does anyone can help me out?
and ...i am really in hurry...thanks...

besides...i need all the steps...
 
Physics news on Phys.org
can somebody help me out?

especially the second question...
 
What have you tried? Have you had any thoughts at all on these problems?
 

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