Abstract Algebra IBL Wiki theorem

theorem:hk_is_a_subgroup_iff_hk_kh

Anonymous (anonymous@undisclosed.example.com) — 2013-08-16T18:08:12+00:00

====== $HK$ is a Subgroup iff $HK=KH$ ====== ====Theorem==== Let $H$ and $K$ be subgroups of a group $G$. Then $HK$ is a subgroup if and only if $HK=KH$. ---- ==== Remarks ==== * See problem [[problem:exploring_the_set_products_hk_and_kh_on_d4]] to have the students conjecture this theorem before having them prove it. ---- ==== $\LaTeX$ version ==== \begin{theorem} Let $H$ and $K$ be subgroups of a group $G$. Then $HK$ is a subgroup if a…

theorem:intersection_of_subgroups_is_a_subgroup

Anonymous (anonymous@undisclosed.example.com) — 2013-08-22T16:09:24+00:00

Intersection Of Subgroups Is A Subgroup Theorem If is a group, then the intersection of any collection of subgroups of is also subgroup. ---------- Remarks * None. ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % None. %%%%% \begin{theorem} If $G$ is a group, then the intersection of any collection of subgroups of $G$ is also subgroup. \end{theorem}

theorem:sylow_s_theorem

Anonymous (anonymous@undisclosed.example.com) — 2013-08-13T11:29:21+00:00

Sylow's theorem Theorem. Let be a finite group and a prime. Write with . Then * acts transitively on by conjugation with modulo , and * every has order . ---------- Remarks * denotes the greatest common divisor of and . * denotes the collection of Sylow $p$-subgroups of . ----------

theorem:template

Anonymous (anonymous@undisclosed.example.com) — 2013-08-15T12:42:10+00:00

@!!PAGE@ Theorem Type the theorem using wiki syntax. ---------- Remarks * Put them in a bulleted list. ---------- $\LaTeX$ version %%%%%%%%%% % DEPENDENCIES % RequiredMacros: \DeclareMathOperator{\syl}{Syl} %%%%%%%%%% \begin{theorem} Type the theorem using LaTeX syntax. \end{theorem}

theorem:the_composition_of_permutations_is_a_permutation

Anonymous (anonymous@undisclosed.example.com) — 2013-11-21T10:01:48+00:00

The Composition Of Permutations Is A Permutation Theorem Let be a set. Suppose that are all Permutation of . Then the composition is a permutation of . ---------- Remarks * Put them in a bulleted list. ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % RequiredMacros: \DeclareMathOperator{\syl}{Syl} %%%%% \begin{theorem} Let $X$ be a set. Suppose that $\sigma_1, \sigma_2, \ldots,\sigma_k$ are all [[Definition.Permutation|permutations]] of $X$. Then the composition $\sigma_1\c…