Subgroup of given order of an Abelian group

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Discussion Overview

The discussion revolves around the properties of subgroups in Abelian groups, specifically whether the existence of a subgroup of a given order implies the existence of an element of that same order within the group. The scope includes theoretical considerations related to group theory.

Discussion Character

  • Exploratory
  • Debate/contested

Main Points Raised

  • Some participants propose that if there exists a subgroup of an Abelian group of order 'm', it should imply the existence of an element of order 'm' in the group.
  • Others question this implication by considering the case where 'm' equals the order of the group itself.
  • A later reply references an elementary Abelian 2-group, suggesting that specific examples may illustrate the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the existence of a subgroup of order 'm' guarantees an element of the same order in Abelian groups, indicating that multiple competing views remain.

Contextual Notes

The discussion does not resolve the implications of subgroup orders and their relationship to element orders, leaving open questions about specific cases and definitions.

siddhuiitb
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Hey!
We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse?
Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not hold in general for Non-Abelian groups. But what about Abelian groups?
 
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Suppose that were true. what if m = |G|?
 
siddhuiitb said:
Hey!
We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse?
Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not hold in general for Non-Abelian groups. But what about Abelian groups?

Take a look at an elementary Abelian 2 -group of order greater than 2 e.g. Z/2 x Z/2 x Z/2.
 
Thanks!:smile:
 

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