• the Blitzkrieg (or lightening war) is the way to fight.  WWI dragged on for millions of soldier deaths with out it.  The Desert storm war was won in 5 days by perfecting the technique.
• It was stupid hubris for both the Germans and Japanese to try to control so much hostile territory.
• The Italians totally sucked in battle, the Germans were always bailing them out. (they must not have been into the whole world conquest thing)
• A fixed defense is a lost cause.  The best defense is actually to fall back and give up ground if you have it.  (The Russians broke the German invasion by falling back a few thousand miles and stretching the German supply lines too thin).  The German losses in the East were largely because of ill-fated arrogant attempts to hold ground.
• Air power means everything.  Since WWII, the victors have always had air supremacy.

• The Turkish slaughter of about 1 million Armenian innocents inspired Hitler.  He was quoted as saying, "No one remembers the million Armenians killed, who cares how we deal with the Jewish problem".
• Much of the bad feelings that led to Hitler rising to power was a result of the unfair Treaty of Versailles which stripped Germany of dignity after WWI.  Many historians now consider both wars as part of the same conflict. It's best to end a war as friends.
• "True Believers" and irrationality are the greatest threats to humanity:  The Nazis were true believers, they really believed that they had to stop an evil conspiracy by the Jews and usher in a golden new age for humanity.
• It's important to note that 2 democracies have never fought.  Countries with a high respect for basic human rights are rarely aggressors.

• it was really the Russians who beat the Nazis (the Nazis had far more troops opposing them and the Russians had almost 10 times the losses as  the Americans).  The communists didn't care how many losses they incurred.
• Germans got very useful experience helping the victors in the earlier Spanish civil war.
• German POWs were kept in Russia till the mid 50's as slave labor and over 75% died in captivity.
• The Soviets under Stalin and the Red Chinese under Mao killed far more people than the Nazis.
• The Swiss sent Jews back to be gassed - or taxed them for being Jewish, they also sold military supplies to the Nazis and kept Jewish assets, in addition to not joining the free world in the fight against totalitarianism.
• The Spanish and Turks didn't fight but leaned in favor of Germany.
• Nazism was occult in origin and opposed to Christianity.
• Anti-Jewish hatred was inspired by phony documents purporting a massive Jewish conspiracy to subvert the world.  These same forged documents are still used by neo-nazis to inspire hatred in our generation.
• The Germans were actually welcomed in much of North Africa because they were kicking out the British who had colonized the region and they were at first welcomed for expelling the Stalinists from Eastern Europe.  Likewise, the Japanese claimed themselves to be liberating Asia from the "pale-skinned round eyes".
• The Americans did inflict a few extra million German civilian casualties by fire bombing many cities and non-military targets.
• The Nazis got many gains via well thought out diplomacy:  they did a lot of bluffing, convinced countries to give up with out a fight, they stole military equipment and money from vanquished countries, they played up hatreds between ethnic factions in conquered lands.
• Stalin invaded Poland along with the Nazis. He also mistreated other conquered countries so badly that they welcomed the Germans in as liberators.
• Although the war brought unprecedented human suffering, it did bring about a number of positive things:
• A chance to relayout a number of vanquished cities.
• The advent of the rocket age (from the German V2 effort to destroy London)
• Computers (via American efforts to crack code and calculate trajectories)
• The American interstate road system (American generals were inspired by the German autobahn)
•  jet aircraft (the Germans first developed them to defend against American bombers).

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As you point out, the Wehrmacht was defeated by the Russian Army -- if Hitler had not invaded Russia, but had invaded Egypt, then Saudi Arabia, avoided his stupid racial policies, and offered reasonable peace terms to Britain, we might be living in a very different world today..