I found that the root.sh script would set the permissions.
I located the root.sh in two locations and ran them both.
/u01/app/oracle/product/Ora10gAS_Inf/root.sh
/u01/app/oracle/product/Ora10gAS_Mid/root.sh
I then ran
"find /u01 -perm +6000 -type f -exec ls -ld {} \;"
and got this result:
-rwsr-s--- 1 root dba 18462 May 15 2007 /u01/app/oracle/product/Ora10.2/bin/nmb
-rwsr-s--- 1 root dba 19656 May 15 2007 /u01/app/oracle/product/Ora10.2/bin/nmo
-rwsr-s--- 1 oracle dba 74794 May 16 2007 /u01/app/oracle/product/Ora10.2/bin/nmupm
-rwsr-s--x 1 oracle dba 901 May 16 2007 /u01/app/oracle/product/Ora10.2/bin/emtgtctl
-rwsr-s--- 1 root dba 13411216 Oct 15 2006 /u01/app/oracle/product/Ora10gAS_Inf/Apache/Apache/bin/ssomigrate
-rwsr-s--- 1 root dba 13302 May 21 2007 /u01/app/oracle/product/Ora10gAS_Inf/bin/nmb
-rws--x--- 1 oracle dba 1805775 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Inf/bin/oidrepld
-rws--x--- 1 root dba 1712084 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Inf/bin/oidldapd
-rws--x--- 1 oracle dba 1773596 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Inf/bin/oidmon
-rwsr-s--x 1 oracle dba 916 May 21 2007 /u01/app/oracle/product/Ora10gAS_Inf/bin/emtgtctl
-rwsr-s--- 1 root dba 13720 May 21 2007 /u01/app/oracle/product/Ora10gAS_Inf/bin/nmo
-rwsr-s--- 1 oracle dba 28071 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Inf/bin/nmupm
-rwSr----- 1 oracle dba 1536 Feb 6 2008 /u01/app/oracle/product/Ora10gAS_Inf/dbs/orapworcl
-rwsr-s--- 1 oracle dba 23219 May 21 2007 /u01/app/oracle/product/Ora10gAS_Inf/opmn/bin/opmnctl
-rwsr-s--- 1 root dba 13411216 Oct 15 2006 /u01/app/oracle/product/Ora10gAS_Mid/Apache/Apache/bin/ssomigrate
-rwsr-s--- 1 root dba 13302 May 21 2007 /u01/app/oracle/product/Ora10gAS_Mid/bin/nmb
-rwsr-s--- 1 root dba 13720 May 21 2007 /u01/app/oracle/product/Ora10gAS_Mid/bin/nmo
-rwsr-s--- 1 oracle dba 28071 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Mid/bin/nmupm
-rwsr-s--x 1 oracle dba 916 May 21 2007 /u01/app/oracle/product/Ora10gAS_Mid/bin/emtgtctl
-rwsr-s--- 1 oracle dba 23219 May 21 2007 /u01/app/oracle/product/Ora10gAS_Mid/opmn/bin/opmnctl
-rwsr-x--- 1 oracle dba 17495 Jan 24 2008 /u01/app/oracle/product/Ora10gAS_Mid/webcache/bin/webcachectl
Once these permissions got set and I installed all the packages required by Oracle, I was up and running again.
I have found my own answer and I have documented it for future use.