I'm not sure how your friend has reached that conclusion. "Viking" wasn't really a term that was much in contemporary use, but is more of a generic modern term to encompass "Early Medieval Scandinavian". In the 9th Century English poem Widsiđ, there is a reference to Wicinga cynn which suggests that, at least informally, "Viking" was used as a synonym for the Scandinavians, but contemporary English sources tend to refer to them simply as "The Danes", "The Heathens", "The Pagans" or just "The Enemy". It's only later in the 10th Century that English sources start to differentiate between separate Scandinavian polities. So to that extent, it's entirely valid to refer to the Danes as a subset of "The Vikings" insofar as that is a modern term used to denote all of Early Medieval Scandinavia rather than a specific subset or groups.

The Great Army which occupied Northumbria and East Anglia in the 9th Century was Danish, or was at the very least identified as such by contemporary English sources, although it's doubtful the extent to which it operated under any kind of centralised royal authority. In contrast, the army which invaded England in the early 1000s under Sweyn Haraldsson and later Cnut was led directly by Kings of Denmark who actively sought to incorporate a unified England into a wider Danish Empire which also incorporated Norway.

William of Normandy was indeed a descendant of Rollo I who was, indeed, a "Viking", again using that as a generic term for Early Medieval Scandinavians. It's worth noting, however, that William and Rollo are separated by 5 generations and over a century, and the Normans of 1066 are Francophone Christians who very much fight in the continental style, and indeed invade England as leaders of a pan-Frankish coalition of soldiery from areas including Brittany, Picardy and the Occitan. While 'ethnically' Scandinavian, therefore, to call William I a "Viking" would be the equivalent of a modern American claiming they are "Irish" because of a single ancestor who crossed the Atlantic in the 1890s.

Vikings is a generic word for Scandinavian warriors from Norway, Sweden and Denmark who operated in the middle medieval era - ca 8th to 11th Centuries and who fought, traded and colonised northern Europe and into Greenland and even into southern Europe. Britain was continually invaded/raided by several of these groups in the era and there was some loose co-operation between these groups.

William the Conqueror though was a descendant of Rollo, a Scandinavan Viking warlord of the 9th century who's birth place and therefore nationality is unknown. He could have been born in modern day Denmark or Norway and there have been claims made for both countries, as far as I know there has been nothing has been definitively proven.

The term "Norman" is a corrution of the word "Northman" suggesting that in France at least they didn't really distinguish between different tribes of Vikings and it likely in military campaigns that Viking armies were formed from warriors from across Scandinavia rather than predominatly one part of it. Having campaigned in northern France Rollo was bought off with lands in the area around Cherbourg and this eventually became Normany and was extended out by Rollo and his descendants, Rollo became the 1st Duke of Normany and William the Conqueror was the 7th.

The Danes were a subset of the Vikings, whether Wiliam was, via Rollo, Danish is not proven, whether it can be claimed that Wiliam was Danish because he had a great, great etc grandfather who was Danish is though a bit of a stretch. By then his branch were well established in Normandy and were Christian rather than pagan etc. However there were plenty of Danes who invaded Britain/England in that era and a fair few of our Kings before William were Danish. The Jutes (from Jutland) were one of the tribes who raided, settled and traded with what is now eastern England and a lot of their dna remains, both literally and culturally/socially in place names, Kent for example.

I'm a little bit confused by the framing of this question. When you say, "it was the Danes that conquered England, not the Vikings," are you referring to the Norman conquest of 1066? Or the conquests by "Vikings" in the 8th through early 11th centuries of Anglo-Saxon England?

I think that u/Jay_CD answered your question about the Norman conquest, and William the Conquerer being descended from Scandinavian Vikings.

But to clarify a little about the earlier period of conquest in England:

Groups of raiders began targeting British sites, starting most infamously with the Raid at Lindisfarne in 793. From that point, until 1066 and even a little beyond, raiders, colonists, and settlers from Norway, Sweden, and Denmark were pretty much a constant presence in the Anglo-Saxon Kingdoms, as well as in what would become Scotland, the Hebrides, the Shetlands, etc, and Ireland.

Some of these groups were from Denmark, some of these groups were from Norway and Sweden. It seems that the "Norse" vikings were most prevalent in the North and West - establishing bases in Northumbria, Scotland, and Ireland. From the perspective of the Anglo-Saxon chroniclers who detailed these events and discussed the raiders and warriors coming to their lands - most of these groups were referred to as "Danes."

The Danish and Norse raiders would generally not have been referred to as "Vikings" - not by themselves, and likely not by the people they were raiding. The etymology of "Viking" is a little hazy, but it seems relatively certain that it was used as an active verb, NOT a racial or ethnic descriptor, so: "They went viking," meaning "they went traveling with the purpose of raiding."

So to say, "was it the Vikings or the Danes who invaded England?" is kind of the wrong question. Some of the Danes who came to England in the period between about 800 and 1066 came as raiders - vikings. Some just came as settlers, looking to move into the land that these Danish warriors had won in East Anglia, Northumbria, and the Eastern part of Mercia - the Danelaw.

The Danes (and Norse) that did come were not a unified political identity - they did not all come under the banner of a single King who was claiming England in his name - for most of this period, England did not exist in the way it does now, but still functioned as several separate and often warring Kingdoms. So there were Danish kings of East Anglia, Danish kings of Jorvik (The Kingdom of York in Northumbria), Norse kings of Jorvik, Norse and Danish warlords who claimed parts and pieces of the "Five Boroughs" - parts of the Danelaw that once belonged to Mercia before the Danish invasions. But all of these peoples were not necessarily united or identified with each other.

The closest thing to a Danish conquest that occurred during this period was the reign of the Danish King Cnut the Great, after the death of Aethelred the Unready in 1016. By 1016, England was a unified Kingdom, and Cnut ruled it as a Christian Dane, as part of his "North Sea Empire," which also included Denmark and Norway.

By 1016, many of the elements that are characteristic of the "Viking age" were starting to change - Scandinavian tribes were consolidating under larger and more powerful kingships, and Scandinavian leaders were taking up Christianity more readily. In this sense, it's not really accurate to describe Cnut and his warriors as "Vikings" - they were not raiders, but were rather a large and organized army participating in the conquest games of Christian kings - much different than pseudo-independent raiding parties made up of "heathens" grabbing the gold from a church because they could.

So, I guess to summarize, I would say:

-"Viking" raiders from mostly Denmark and Norway started grabbing land in Northern and Eastern "England" in the 800s

-This was not a conquest of England because England did not exist yet, but by the late 800s, the "Vikings" - Danes and Norse - did hold power in all of the Anglo-Saxon kingdoms except for Wessex.

-The continued conflict between Anglo-Saxons and "Vikings" resulted in the birth of England as a unified political identity, held by the Anglo-Saxons with many Danish and Norse settlers still living in their lands.

-In 1016, the Dane Cnut took power and ruled the kingdom of England alongside Denmark and Norway. It is a stretch to call Cnut a Viking, but his presence in England was certainly the result of the longstanding relationship between Scandinavia and England that was forged in part by the earlier Viking conquests.