Order of Elements in a Group: A Quick Check of Understanding

  • Context: Graduate 
  • Thread starter Thread starter Artusartos
  • Start date Start date
  • Tags Tags
    Elements Group
Click For Summary
SUMMARY

The discussion centers on the understanding of group theory, specifically regarding a group of order 42 containing the subgroup Z_6. The participant incorrectly assumes that the group of units of Z_6, which has order 2, directly indicates the presence of elements of order 6 in the larger group G. Additionally, the calculation of cyclic subgroups of order 6 based on the normalizer N(P) is also questioned, highlighting that the reasoning presented lacks clarity and correctness.

PREREQUISITES
  • Understanding of group theory concepts, particularly cyclic groups and their orders.
  • Familiarity with the structure of the group of units in modular arithmetic, specifically Z_6.
  • Knowledge of normalizers in group theory and their significance in subgroup analysis.
  • Basic principles of element orders within groups and their implications.
NEXT STEPS
  • Study the properties of cyclic groups and their subgroups in detail.
  • Learn about the group of units in modular arithmetic, focusing on Z_n for various n.
  • Explore the concept of normalizers in group theory and their role in subgroup classification.
  • Investigate the implications of element orders in finite groups and their relationships.
USEFUL FOR

This discussion is beneficial for students and enthusiasts of abstract algebra, particularly those studying group theory, as well as educators seeking to clarify concepts related to cyclic groups and their properties.

Artusartos
Messages
236
Reaction score
0
I just want to check if there is anything wrong with my understanding...

Let's say we have a group of order 42 that contains Z_6. Since the group of units of Z_6 has order (3-1)(2-1), it means that we have 2 elements of order 6 in G, right? In other words, for any cyclic subgroup of order n, we just calculate the group of units to see how many elements of order n we have. Is that correct? Also, if we let P=Z_6, we can have at most 7 cyclic subgroups of order 6, since 7=42/|N(P)| (where N(P) is the normalizer of P), if N(P)=6 (which is the smallest it can be, since it has to contain Z_6). I'm just wondering if my understanding is correct.

Thanks in advance
 
Physics news on Phys.org
Not right. We might have elements of order ##6## in ##G## or not, but your reasoning with the units of ##\mathbb{Z}_6## is strange. The non-units you might refer to are ##0## and ##3## and none of which has order ##6##.
 

Similar threads

Undergrad Localising a non integral domain at a prime
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
MHB Is Every Group of Order 25 Cyclic?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How to find group types for a particular order?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
1K
MHB Question on subgroup and order of the elements
  • · Replies 7 ·
Replies
7
Views
2K
MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Is G simple?Is G simple? A different approach using Sylow theorems
  • · Replies 5 ·
Replies
5
Views
3K
MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad First Sylow Theorem: Group of Order ##p^k## & Cyclic Groups
  • · Replies 1 ·
Replies
1
Views
2K