Jordan-Holder ©w²z

Let G be a finite group. If G is not simple, then G has a normal subgroup #tex2html_wrap_inline163# such that #tex2html_wrap_inline165# is simple<#20#>.<#20#> Just let #tex2html_wrap_inline167# be a maximal normal subgroup in the finite group G<#21#>.<#21#> Similarly, #tex2html_wrap_inline171# has a normal subgroup #tex2html_wrap_inline173# such that #tex2html_wrap_inline175# is simple. Thus we get a descending chain

#displaymath143#

such that #tex2html_wrap_inline177# is normal in #tex2html_wrap_inline179#, and #tex2html_wrap_inline181# is simple. Now suppose that

#displaymath144#

is another such chain of subgroups. The Jordan-Holder Theorem asserts that m=n, and that there is a one-to-one correspondence between the factor groups #tex2html_wrap_inline185# and #tex2html_wrap_inline187# such that corresponding factor groups are isomorphic. We will prove this remarkable theorem, but first some notation and terminology are needed. <#26#>The groups considered are not necessarily finite.<#26#>

#definition27#

#theorem32#

\ \ If a group has a composition series of length one, then the group is simple and any two composition series are certainly equivalent. Now suppose that a group G has a composition series

#equation34#

of length n;SPMgt;1, and that if a group has a composition series of length less than n, then any two composition series of that group are equivalent. Let

#equation37#

be any composition series of G. Consider the series

#equation40#

and

#equation43#

Since #tex2html_wrap_inline215# is a normal subgroup of #tex2html_wrap_inline217# and #tex2html_wrap_inline219#, the Third Isomorphism Theorem (<#48#>2.3.12<#48#>) yields

#displaymath146#

#displaymath147#

and #tex2html_wrap_inline221# is a normal subgroup of #tex2html_wrap_inline223# since it is a product of two normal subgroups. Since #tex2html_wrap_inline225# is a simple group, #tex2html_wrap_inline227# is either #tex2html_wrap_inline229# or #tex2html_wrap_inline231#. That is, #tex2html_wrap_inline233# is either #tex2html_wrap_inline235# or #tex2html_wrap_inline237#. Therefore, if we remove repetitions from

#displaymath148#

we get a composition series for #tex2html_wrap_inline239#. By our induction hypothesis, the resulting composition series is equivalent to the composition series

#displaymath149#

and hence (#one#63>) and (#three#64>) (with repetitions removed) are equivalent. #tex2html_wrap_inline241#