This is a deep-seated institutional problem that pervaded the Soviet military from the 1960s onwards, and continued into the Russian military well beyond the 20 year rule, although can't get into that of course.

This phenomenon is known as dedovshchina, or "rule of the grandfathers", and is essentially institutionalized hazing on an absolutely brutal level. The basic structure revolved around the conscription schedule of the Soviet military introduced in the mid-'60s, which brought in new recruits every six months to serve 2 year hitches in the Red Army. The soldiers in the last year of their service were known as the stariki or deda, the proverbial 'grandfathers', and they ruled the roost of the barracks. The target of their abuse was the newest group of recruits who, for their first six months, thus were forced to endure a litany of abuse, the weakest of the group referred to as "“sestra” or sisters, and the worst abused, although none of the 'sons' expected to escape unscathed. The deda broke into two groups, those in their last six months, who were less physically abusive and instead relied on the new recruits for menial labor, and those in their 'third quarter' who were considered by far the most vicious.

The actual, physical mistreatment which you touch on was perhaps the worst, but these new recruits were also expected to work as the personal servants of the 'grandfathers' and do any menial duties on base that the veterans were supposed to be doing themselves, and were generally victims of theft. Money from home would rarely last a few minutes once a deda caught wind. Further, due to lack of supplies, new soldiers would invariably have their clean, new uniforms requisitioned by the veterans and provided with the old rags that their tormentors had been wearing. Although not common, cases of rape were not unknown. Once the newer draft class came in, they were 'off the hook' - although if they made themselves targets they might continue to be harrassed - and then they in turn would 'pay it forward', perpetuating the cycle of violence and abuse.

To understand why this was allowed, in the first, the short time of service itself contributed, as this never allowed an esprit de corps to develop within the enlisted ranks. The officers were distant from the men, rarely forming any sort of actual relationship with their men. Some would plead ignorance of the dynamic, while others just didn't care. Additionally, keep in mind that the Red Army lacked an institutional corps of NCOs, the sergeants being conscripts on the same schedule who simply stood out in training. In fact the sergeants likely had suffered under the 'grandfathers' in their first six months, and rather then reining it in, were more likely hoping to have their turn on the top rather than trying to stop it. In any case, while the sergeants might have held de jure status within the military hierarchy, that took a backseat inside the barracks where the seniority of the 'grandfathers' was simply more important.

There was some variation in how bad it was, and memoirs and records show that it correlated strongly with the level of training a unit received, and how efficiently they were resupplied. Well tended units were less inclined to support such abuse, or at least the worst of it, while the more ignored units would see the worst kind of excesses. Construction units, considered the dumping grounds for the worst draftees had a reputation for seeing the worst hazing. Beyond even that though ethnic minorities perhaps had it the worst of all, often shunted into the worst units anyways, but additionally subjected to 'special treatment' by ethnic Russians, such as one Chechen who recalled "They cursed me as a savage, a traitor, a bandit, what have you." Desertion, and even suicide, often could be the result.

This was, as I said, well institutionalized in the military, and given the scope of conscription, it was widely known by those who had been in the Army, but not widely discussed. It was only in the Glasnot era that it was publicly acknowledged as a problem. Publications like Pravda would publish accounts, such as this one which is cited by Reese:

The sergeant always beat me and the other new recruits. One time he told us to write home and ask them to send fifty rubles. He took the money away from those who received it. I complained about the beatings, and I was called in to see the commander. He listened and said, “Don’t write any more letters anywhere; the sergeant won’t lay a hand on you.” That very day he thrashed me good.

And Iurii Poliakov's novella "One Hundred Days Before the Command" was published in 1987, additionally throwing open the doors on such ill-treatment and defining the dynamic thusly:

If we ‘sons’ wish to have an uneventful service and a happy homecoming, we have to obey the ‘grandfathers’ and live according to the untouchable barrack laws. The army is not the same as civilian life … the army has its own laws: you are screwed in your first year and then you screw in your second.

This public reckoning in the '80s, combined with the demoralizing effects of the Afghanistan mire (where dedovshchina by the veterans with the new arrivals could be quite bad even by the standards of the Red Army) was a major blight on the prestige of the Red Army, and there was open advocacy exerted on the authorities, by groups such as "The Committee of Soldiers' Mothers", but whatever impact they had on reform, it certainly didn't eliminate the problem. The collapse of the Soviet Union upended the embryonic reform movement, and within the Army itself, saw dedovshchina worsen, a reaction of the internal military culture to the instability of the times and the corruption of the Russian Armed Forces. Further attempts at reform have been attempted with varying degrees of success in recent years, but that is on the wrong side of the 20 year rule, unfortunately.

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Sources:

The Soviet Military Experience: A History by Roger R. Reese

Red Commanders: A Social History of the Soviet Army Officer Corps by Roger R. Reese

Alena Maklak (2015) Dedovshchina on trial. Some evidence concerning the last Soviet generation of “sons” and “grandfathers”, Nationalities Papers, 43:5, 682-699

Dale R. Herspring (2005) Dedovshchina in the Russian Army: The Problem That Won't Go Away, Journal of Slavic Military Studies, 18:4, 607-629

I can't really get into recent attempts at reform as that is on the wrong side of the 20 year rule, but check out:

Daucé, Françoise. 2014. "Dedovshchina After the Reform." Problems Of Post-Communism 61, no. 2: 36-45. International Security & Counter Terrorism Reference Center