Does Order Matter in Cyclic Subgroups/Groups?

  • Thread starter Thread starter Gear300
  • Start date Start date
  • Tags Tags
    Cyclic
Gear300
Messages
1,209
Reaction score
9
Since certain operations are not commutative, when a group G = <a, b>, does the order matter (so that <a, b> is not necessarily equal to <b, a>)?
 
Mathematics news on Phys.org
Assuming that by <a,b> you meant "the group spanned by a and b", the order does not matter. In that case the notation <a, b> -- as well as <b, a> -- means: take elements a and b and multiply them and their inverses until you get a group. Actually, it means: take the smallest group of which a and b are elements. In particular this means that both ab and ba must be elements of both <a, b> and <b, a>. Actually, a more concise notation would be: <{a, b}>
 
except in this case we say group 'generated' by a and b. for some reason 'spanned' is used only for vector spaces and (some times) modules.
 
I see...thanks for the replies.
 
Whoops, thanks nirax. Was already wondering why that sounded so odd ;)
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'Useless continued fraction for 1'

I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
View full post »

Similar threads

POTW Groups of Prime Power Order
Replies
1
Views
1K
MHB Is Every Group of Order 25 Cyclic?
Replies
1
Views
1K
Characterizing subgroups of a cyclic group
Replies
9
Views
1K
I Subgroup axioms for a symmetric group
Replies
1
Views
1K
POTW Product of Two Finite Cyclic Groups
Replies
1
Views
2K
What is a Commutator Subgroup?
Replies
1
Views
2K
Cyclic group has 3 subgroups, what is the order of G
Replies
5
Views
2K
What is the Definition and Explanation of a Quotient Group?
Replies
1
Views
2K
Group is a union of proper subgroups iff. it is non-cyclic
Replies
1
Views
4K
Every infinite cyclic group has non-trivial proper subgroups
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top