Faced with evidence of widespread cheating to boost enlistments, the Army today announced it will expand its investigation of fraud to its entire recruiting command.

Army investigators will check records and visit all 57 recruiting districts in the country by Oct. 31, according to Maj. Jeff Cook, an Army spokesman at the Pentagon. A report on the fraud and recommendations to eliminate it is due by mid-November, he said.

So far the Army has relieved 165 recruiters, including three officers, in 17 recruiting districts covering parts of 25 states since the probe began five months ago. The Army has 6,372 recruiters and supervisory staff nationwide.

"It's now just coming to a head," Cook said. "We're going to start having some rapid changes in the next couple of days."

Hundreds, perhaps thousands, of recruits were illegally enlisted in the Army as a result of the recruiting fraud, officials have said.

Cook said 20 applicants in a delayed entry program were discharged as a result of the investigation but the full extent of the fraud will not be outlined until the report is prepared.

Investigators have focused on allegations that recruiters illegally coached applicants on entrance exams, passed out bootleg copies of the exams and altered records to enable unqualified applicants to enlist.

The investation began in Charlotte in May and spread to other districts in June. Cook said the "overall idea from the beginning" was to investigate the entire recruiting command, but that officials did not want to "broadcast" the extent of the investigation by disclosing it earlier.

In interviews, some Charlotte district recruiters admitted they gave out bootleg copies of the entrance tests, held "schooling" sessions in recruiting stations, faked high school records and birth certificates, forged signatures, passed out counterfeit Social Security cards, ignored police records, lied and told applicants to lie.

The recruiters said the cheating was the result of pressure to meet enlistment quotas.Those who met quotas could win awards, earn up to $150 a month extra pay and get faster promotions. Those who failed could be transferred with damaging efficiency reports.

Congressional opponents to the all volunteer Army have said the cheating reflects the difficulties in attracting qualified soldiers. The Army just finished its worst recruiting year since the draft ended in 1973.

Supporters of the volunteer force have said the fraud shows entrance standards are too high. The Army has lowered educational requirements to enlistees since the probe began.

Cook said a 54-member investigating team, led by deputy commander of recruiting Brig. Gen. Donald Conelly, has already visited the worst districts believed to have the most extensive fraud, including Charlotte and Montgomery, Ala.

The Army suspended 32 of 55 recruiters in Charlotte, including the commanding officer, and suspended 41 of 74 recruiters in Montgomery, including the commanding officer and the operations officer.

Courts-martial are scheduled to begin Oct. 23 at Fort Bragg, N.c., for three Charlotte district sergeants charged with sponsoring fraudulent enlistments and falsifying documents. Six other Charlotte noncommissioned officers also face court-martial charges, while four sergeants face less serious charges.

Cook said the Army probe has relied heavily on a Pentagon study of Army, Navy, Marine and Air Force recruiting records from October 1978 to June 1979. The study indicated 1,125 of 14,442 recruiters in the four services may have coached applicants on entrance tests.