Show <a in D8 : a^2=1> is not a group

Thread starter Daniiel
Start date Mar 18, 2012
Tags

Group

Click For Summary

SUMMARY

The discussion centers on demonstrating that the set does not form a group. Participants clarify that the interpretation of corresponds to the dihedral group D4, which is defined by the relations <(a,b): a^2=e, b^2=2, ab=a^{-1}b>. The correct interpretation involves elements of D8, specifically identifying elements of order 2 and confirming that the product of two such elements does not yield another element of order 2, thus violating group properties.

PREREQUISITES

Understanding of group theory and its axioms
Familiarity with dihedral groups, specifically D4 and D8
Knowledge of group element orders and their implications
Basic algebraic manipulation of group relations

NEXT STEPS

Study the properties of dihedral groups, focusing on D8 and its elements
Learn about group orders and how they affect group structure
Explore examples of non-group structures in algebra
Investigate the implications of the group axioms in various mathematical contexts

USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying group theory, particularly those interested in dihedral groups and their properties.

Mar 18, 2012

Daniiel

Hey,

I've been trying to solve this question,

Show that <a in D8 : a²=1> is a not a group.

I might not be processing it properly, but my interpretation of the question is that

<a in D8 : a²=1> = <(a,b): a²=e, b²=2, ab=a^-1b>

Which is just D4, a group, the set of dihedral where 2n=4,

Is the proper interpretation supposed to be

<a in D8 : a²=1> = <(a,b): a²=e, a⁴=e,b²=2, ab=a^-1b>

which is again D4,

Could anyone help me interoperate the question

Physics news on Phys.org

Mar 18, 2012

jgens

Gold Member

Notice that D₈ = {e,r,r²,r³,s,sr,sr²,sr³}. Now check which of these elements have order 2. Then find two elements of order 2 which when multiplied together do not produce an element of order 2.