Vipul: /* Related facts */

2010-01-22T00:46:12Z

Related facts

← Older revision		Revision as of 00:46, 22 January 2010
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	* [[Cyclic normal implies hereditarily normal]]		* [[Cyclic normal implies hereditarily normal]]
	* [[Hereditarily characteristic implies abelian]]
	* [[Hereditarily characteristic not implies cyclic in finite]]		* [[Hereditarily characteristic not implies cyclic in finite]]
	* [[SQ-dual::Cyclic-quotient characteristic implies upward-closed characteristic]]		* [[SQ-dual::Cyclic-quotient characteristic implies upward-closed characteristic]]

Vipul: Created page with '{{subgroup property implication| stronger = cyclic characteristic subgroup| weaker = hereditarily characteristic subgroup}} ==Statement== Suppose $G$ is a group …'

2010-01-22T00:37:34Z

Created page with '{{subgroup property implication| stronger = cyclic characteristic subgroup| weaker = hereditarily characteristic subgroup}} ==Statement== Suppose <math>G</math> is a group …'

New page

{{subgroup property implication|
stronger = cyclic characteristic subgroup|
weaker = hereditarily characteristic subgroup}}

==Statement==

Suppose <math>G</math> is a [[group]] and <math>K</math> is a [[cyclic characteristic subgroup]] of <math>G</math>, i.e., <math>K</math> is a [[cyclic group]] and is also a [[fact about::characteristic subgroup]] of <math>G</math>. Then, <math>K</math> is also a [[hereditarily characteristic subgroup]] of <math>G</math>, i.e., every subgroup <math>H</math> of <math>K</math> is characteristic in <math>G</math>.

==Related facts==

* [[Cyclic normal implies hereditarily normal]]
* [[Hereditarily characteristic implies abelian]]
* [[Hereditarily characteristic not implies cyclic in finite]]
* [[SQ-dual::Cyclic-quotient characteristic implies upward-closed characteristic]]

==Facts used==

# [[uses::Cyclic implies every subgroup is characteristic]]
# [[uses::Characteristicity is transitive]]: If <math>A \le B \le C</math> with <math>A</math> characteristic in <math>B</math> and <math>B</math> characteristic in <math>C</math>, then <math>A</math> is characteristic in <math>C</math>.

==Proof==

'''Given''': A group <math>G</math> with a cyclic characteristic subgroup <math>K</math>. A subgroup <math>H</math> of <math>K</math>.

'''To prove''': <math>H</math> is characteristic in <math>K</math>.

'''Proof''':

# <math>H</math> is characteristic in <math>K</math>: This follows from fact (1).
# <math>H</math> is characteristic in <math>G</math>: This follows from the previous step, the given datum that <math>K</math> is characteristic in <math>G</math>, and fact (2).

Cyclic characteristic implies hereditarily characteristic - Revision history

Vipul: /* Related facts */

Vipul: Created page with '{{subgroup property implication| stronger = cyclic characteristic subgroup| weaker = hereditarily characteristic subgroup}} ==Statement== Suppose G is a group …'

Vipul: Created page with '{{subgroup property implication| stronger = cyclic characteristic subgroup| weaker = hereditarily characteristic subgroup}} ==Statement== Suppose $G$ is a group …'