# Homework Help: Sylow's First Theorem

1. Oct 6, 2012

### doubleaxel195

I'm getting a little confused about what exactly Sylow's first theorem says.

On Wikipedia, it says that Sylow's First Theorem says "For any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order p^n."

Then in the section of the proof it is restated as "A finite group G whose order |G| is divisible by a prime power p^k has a subgroup of order p^k."

To me, these seem like they are saying two different things. The first seems like its only guaranteeing subgroups of prime power where the prime power is maximal (p^k where |G|=p^km where p does not divide m), while the second seems to be saying there are subgroups of order p^1, p^2, p^3, ... p^(k-1), p^k. Am I misunderstanding something?

2. Oct 6, 2012

### Robert1986

The book has a slightly stronger statement as Sylow's Theorem than what the wikipedia article says. However, if P is a p-group of order p^k then there is a subgroup of order p^j for 0 <= j <= k. To prove this, you can use induction and the fact that p-groups have non-trivial centers and the fact that abelian groups (you can only use abelian properties for the center) have subgroups of order p for every prime that divides the order of the group (there is probably an exercise or this is a theorem where this fact is proven).

3. Oct 6, 2012

### jdinatale

Sylow's firs theorem is basically saying that for any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order p^n.
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9156323075294495, "perplexity": 360.67756742923353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267859817.15/warc/CC-MAIN-20180617213237-20180617233237-00073.warc.gz"}