Cauchy's theorem (group theory)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Group_theory/NfowcI5H">group theory</a>, Cauchy's theorem states that if G is a <a href="/facts/Finite_group/vv2tvogB">finite group</a> and p is a <a href="/facts/Prime_number/L20FLkgn">prime number</a> dividing the <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive <a href="/facts/Integer/PUwwrawx">integer</a> with xp = e, where e is the <a href="/facts/Identity_element/9nss3xHY">identity element</a> of G. It is named after <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a>, who discovered it in 1845.
The theorem is a partial converse to <a href="/facts/Lagrange%2527s_theorem_(group_theory)/VgyNjGlz">Lagrange's theorem</a>, which states that the order of any <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of a finite group G divides the order of G. In general, not every divisor of 
 
 
 
 
 |
 
 G
 
 |
 
 
 
 {\displaystyle |G|}
 
 arises as the order of a subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
. Cauchy's theorem states that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> <a href="/facts/Generator_(mathematics)/Jo1eBRbD">generated</a> by the element in Cauchy's theorem.
Cauchy's theorem is generalized by <a href="/facts/Sylow_theorems/aVTxDRUo">Sylow's first theorem</a>, which implies that if pn is the maximal power of p dividing the order of G, then G has a subgroup of order pn (and using the fact that a p-group is <a href="/facts/Solvable_group/Na7lKIns">solvable</a>, one can show that G has subgroups of order pr for any r less than or equal to n).

Cauchy's theorem (group theory) open-in-new

Cauchy's theorem (group theory)